Illuminations: Linking Length, Perimeter, Area, and Volume

Linking Length, Perimeter, Area, and Volume

Purple Prisms

This Internet Mathematics Excursion is based on E-example 6.3.2 from the NCTM Principles and Standards for School Mathematics. This is the last activity in a sequence of four lessons designed for students to understand scale factor and surface area of various rectangular prisms. Students manipulate the scale factor that links two three-dimensional rectangular prisms to learn about edge lengths and surface area relationships.

Learning Objectives
 
Students will:
  • explain surface area of a rectangular prism
  • describe the relationship between the scale factor and the surface area of two similar rectangular prisms
  • use the applet graph to explain the relationship between the scale factor and the surface area of two similar rectangular prisms
  • create pattern units of squares, predict how patterns with different numbers of squares will appear when repeated in a grid, and check their predictions
  • analyze how repeating patterns are generated

Materials
 
Side Length, Volume, and Surface Area of Similar Solids Applet
Purple Prisms Activity Sheet
Calculators

Instructional Plan

Engage students in a class discussion about their knowledge of surface area, making certain they understand surface area is the two dimensional measurement of a three-dimensional figure. Use several different rectangular prisms for students to demonstrate their understanding of surface area.

Guiding Questions:

  • How do you find area of a rectangle?
  • How many rectangular faces make up a rectangular prism?
  • Although each face of the rectangular prism is two-dimensional, together what do they make?
  • What is the sum of the areas of the faces on a rectangular prism called?
  • Why are square units used when measuring surface area?
  • In your own words, give a definition of surface area of a rectangular prism?
Go on-line to the Side Length, Volume, and Surface Area of Similar Solids Applet. When the applet opens, students should click on "Show Surface Area." There are two similar rectangular prisms, one purple and one red. The red prism remains the same. The length is 1.73 units, the width is 1 unit, and height is 1 unit. Students should use the slide bar or the red dot to change the size of the purple rectangular prism.

Next students should change the size of the purple prism and observe the change in the ratio of L : 1 (scale factor). Note how the Surface Area of Prism A and Prism B and their ratio change for different length measures.

Using the Rectangular Prisms table on the Purple Prisms activity sheet, students should record the Surface Area of A, the Surface Area of B, the ratio of Surface Area A : B, and the scale factor L : 1 for ten different rectangular prisms.

Purple Prisms Activity Sheet Purple Prisms Activity Sheet

Ask students the following questions as they complete the activity:

  • Why is Surface Area B always 8.93 square units?
  • How do you know what the width and height of A are?
  • Are these prisms similar? How do you know?
  • Which column from the Student Learning Guide represents scale factor? Why?

Students should now look at the L : 1 and Surface Area A : B ratio on the table. Click on Show Surface Area graph. Use the Questions for Students (below) to conclude the lesson.

Questions for Students
 
  • What is the relationship between L : 1 and Surface Area A : B?
  • Does anyone have a whole number for L : 1 in his or her table? What is the relationship between that whole number and Surface Area A: B?
  • Apply that relationship to the other numbers in the L  : 1 column and Surface Area A : B column?
  • How does the applet graph illustrate this relationship?
  • Is this graph linear? Why or why not?

Assessment Options
 
  1. Individually, students should sketch a rectangular prism, labeling dimensions. Using the sketch, they should describe surface area. Assess students’understanding that surface area is the sum of the six rectangular faces on the prism measured in square units.

  2. Students can explain in writing the relationship between scale factor and the ratio of Surface Area A : B.

NCTM Standards and Expectations
 
Geometry 6-8
  1. Use geometric models to represent and explain numerical and algebraic relationships.
  2. Use visual tools such as networks to represent and solve problems.
  
1 period   

NCTM Resources

Principles and Standards for School Mathematics
National Council of Teachers of Mathematics Thinkfinity Verizon Foundation
More and Better Mathematics for All Students
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