Cube/RectangularPrism Activity

## Teacher Lesson Plan

This activity is aligned to NCTM Standards - Grades 6-8: Geometry, Problem Solving, Reasoning and Proof, and Communication and California Mathematics Standards Grade 7: Measurement and Geometry #2.1, 2.3, 3.4, 3.5 and Mathematical Reasoning #1.1, 2.1, 2.4, 2.5, 2.6

Students manipulate cubes and rectangular prisms, drawing geometric shapes using grid paper and also manipulating geometric shapes using the ESCOT Runner software to observe geometric relations. Students think about the linear measurements and the corresponding area and volume measurements of the cube and rectangular prism. Ideas of scaling are investigated using both a manipulative and technology. Finally, students are asked to respond to questions in a paper/pencil activity.

It is important to build a bridge between the technology representing the shapes, the 3-dimensional objects, and the 2-dimensional drawings of 3-dimensional objects. Visiting the problem using both techniques addresses a variety of learning styles, brings the abstract into the concrete, and offers interaction with the computer as students investigate, discover, form hypotheses, draw conclusions, and benefit from the quick feedback, the interest and the increased visualization of scaling that a computer provides. Once students have had these experiences it is important to arrive at a synthesis by spending the time necessary to internalize the concepts. Manipulatives can provide space for group work, computers can afford individual explorations, and a synthesis can take place during a full-class discussion. Students can then demonstrate their individual understandings through the writing process.

 Using Manipulatives

Introducing the activity:

1. Pass out the following materials for each student:
2. Instruct the students to cut on the lines of the nets being careful to leave the tabs for easier assembly of the solid shape. Have them fold on the lines and tape.
3. Once the three-dimensional shapes have been assembled introduce or review the vocabulary:
cube
rectangular prism
vertex (vertices)
base
surface area
volume

Drawing Geometric Solids:

1. Pass out one sheet of isometric dot paper to each student and instruct them to draw a unit cube. Discuss the idea of why it is called a unit cube.
2. Introduce the idea of a scaling factor for the sides of the cube. Draw another cube using a scaling factor of 2.
3. Here is an example of how the drawings might look.
4. Repeat this process with the rectangular prism.

 Using Technology

ESCOT Runner software:

Nathalie Sinclair wrote the program for the Cube Exploration Activity

Have the students respond to these questions:

1. What is the relation between the side length of the pink cube and the side length of the blue cube?

2. What is the relation between the area of one face of the pink cube and the area of one face of the blue cube? Express this relation algebraically.

3. What is the relation between the volume of the small cube and the volume of the large cube? Can you express this relation algebraically?

4. What happens to the area of the base when you double the length?

5. Explain the concept of "scale factor" with respect to length, base area, surface area and volume of two cubes.

6. What different combinations of scalings result in solid figures with the same surface area? or with the same volume?

Nathalie Sinclair wrote the program for the Rectangular Prism Exploration Activity

Rectangular Prism Activity

Have the students respond to these questions:

1. What is the relation between the side length of the pink rectangular prism and the side length of the blue rectangular prism?

2. What is the relation between the side width of the pink rectangular prism and the side width of the blue rectangular prism?

3. What is the relation between the side height of the pink rectangular prism and the side height of the blue rectangular prism?

4. What is the relation between the area of the base of the pink rectangular prism and the area of the base of the blue rectangular prism? Express this relation algebraically.

5. What is the relation between the front area of the base of the pink rectangular prism and the front area of the base of the blue rectangular prism? Express this relation algebraically.

6. What is the relation between the side area of the base of the pink rectangular prism and the side area of the base of the blue rectangular prism? Express this relation algebraically.

7. What happens to the area of the base when you double the length?

8. What happens to the area of the base when you double the width?

9. What happens to the area of the base when you double the height?

10. What happens to the volume when you double one of the dimensions (length or width or height)?

11. Explain the concept of "scale factor" with respect to length, width, height, base area, front area, side area, surface area and volume of two rectangular prisms.

12. What different combinations of scalings result in solid figures with the same surface area? or with the same volume?

 Synthesizing the Activity

At this point, students have investigated the problem using manipulatives (constructing three-dimensional solids and drawing solids on using isometric dot paper) and technology (the ESCOT Runner problem).

Discuss with the students the formulas to calculate the area and volume of a cube and rectangular prism given the length of the side or the length, height and width of the prism.

Ask them to summarize the information they have learned from the scaling factor application with respect to a cube and also a rectangular prism.

 Extensions/Resources

Three Dimensional Box Applet: Working With Volume - Kristen Carvell
Box Folding: Using the Box Folding Applet - George Reese
Polyhedra in the Classroom
Studying Polyhedra Poly - Kagi Shareware
Defining Geometric Figures - Ask Dr. Math FAQ
Geometric Formulas - Ask Dr. Math FAQ