Module 2 : Exploring Changes in Area and Volume in Middle School Mathematics

Bonsangue, M., Costa, V., and Ambrosetti, D. (2005). Module 2: Exploring Changes in Area and Volume in Middle School Mathematics. In V. Costa, M. Bonsangue, and H. Shultz (Eds.) (2 nd ed.). Professional Development Resources Online for Mathematics [Online Course].  Available from http://www.pdrom.coursepath.org/.

Motivating CAHSEE Problem

A cereal manufacturer needs a box that can hold twice as much cereal as the box shown below. Which of the following changes will result in the desired box? (V = lhw)

1. Double the height only.
2. Double both the length and width.
3. Double both the length and height.
4. Double length, height, and width.

The largest possible circle is to be cut from a 10-foot square board. What will be the approximate area, in square feet, of the remaining board (shaded region)? ( A = Pi( r 2) and Pi » 3.14)

1. 20
2. 30
3. 50
4. 80

California
Mathematics Content Standards

Measurement and Geometry (7th)

• 2.0 Students compute the perimeter, area, and volume of common geometric objects and use the results to find measures of less common objects. They know how perimeter, area, and volume are affected by changes of scale.
• 2.3 Compute the length of the perimeter, the surface area of the faces, and the volume of a three-dimensional object built from rectangular solids. Understand that when the lengths of all dimensions are multiplied by a scale factor, the surface area is multiplied by the square of the scale factor and the volume is multiplied by the cube of the scale factor.

Measurement and Geometry (7 th)

• 2.0 Students compute the perimeter, area, and volume of common geometric objects and use the results to find measures of less common objects. They know how perimeter, area, and volume are affected by changes of scale.
• 2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.

Math Concepts

• Using formulas to find the volumes of prisms and cylinders; and
• Examining how changes in dimensions of a prism result in changes in its volume

• Using formulas to find the area of two-dimensional figures;
• Solving multi-step problems

Instructional Strategies

• Providing physical models (specifically models of prisms with wooden cubes);
• Generating and testing hypotheses to increase student understanding
• Using inductive and deductive reasoning activities to develop students' mathematical understanding
• Developing student oral explanations to increase their conceptual understanding and problem-solving ability

• Using developmentally–appropriate instructional to support and engage students
• Using students' prior knowledge (students' attitudes, experiences, and knowledge) to enhance student learning
• Addressing the visual, auditory, and kinesthetic learning styles of students
• Using cooperative learning in the form of pair-sharing or specific role assignments
• Creating assignments that allow students to use different intelligences while "doing" mathematics