Illuminations: Linking Length, Perimeter, Area, and Volume

Linking Length, Perimeter, Area, and Volume

Fill'r Up

This Internet Mathematics Excursion is based on E-example 6.3.2 from the NCTM Principles and Standards for School Mathematics. This is the third in a sequence of four lessons designed for students to understand scale factor and volume of various rectangular prisms. In this lesson, the student can manipulate the scale factor that links two three-dimensional rectangular prisms and learn about the relationships between edge lengths and volumes.

Learning Objectives
 
Students will:
  • explain the three-dimensionality of volume
  • apply the volume formula for rectangular prisms
  • understand that the relationship between the volume of two similar rectangular prisms and the scale factor is cubic
  • use the applet to explain the relationship between the volume of two similar rectangular prisms and the scale factor
  • 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
Fill'r Up Activity Sheet
Calculators

Instructional Plan

Engage students in a class discussion about their knowledge of volume by using several different rectangular prisms. Build upon the concepts from the two previous lessons in this unit.

Students should open the Side Length, Volume, and Surface Area of Similar Solids Applet. When the applet opens, students should click on "Show Volume." 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 the height is 1 unit. 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). Students should note how the volume of Prism A, the volume of Prism B, and their ratio change.

Using the Rectangular Prisms table on the Fill'r Up activity sheet, students should record Volume A, Volume B, the ratio of Volume A B, and the scale factor L : 1 of ten different rectangular prisms.

Fill'r Up Activity Sheet Fill'r Up Activity Sheet

Guiding Questions to Ask Students:

  • Why is Volume B always 1.73 units?
  • How do you know what the width, depth, and height of A are?
  • Are these prisms similar? How do you know?
  • Which column in the Rectangular Prisms table on the Student Learning Guide represents the scale factor? Why?

Students should now choose a number from the Volume A column. Using a calculator and the other necessary numbers in that row on the table, calculate Volume A. Repeat this process with other volumes listed on the table.

Look at the L : 1 and Volume A: B ratio. Ask students the following questions:

  • What is the relationship between L : 1 and Volume 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 Volume A : B?
  • Apply that relationship to the other numbers in the L : 1 column and Volume A : B column?
  • How does the applet graph illustrate this relationship?
  • Is this graph linear? Why or why not?

Use students' responses as a way of assessing their understanding of the lesson's concepts.

Assessment Options
 
  1. In pairs, students can discuss why volume is a cubic measurement. In their disucssion, they should give examples of the uses of volume and perhaps draw some sketches to explain their answers.

  2. Once again in pairs, students can explain the relationship between scale factor and the ratio of Volume A : B.

NCTM Standards and Expectations
 
Geometry 6-8
  1. Precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties.
  2. Use geometric models to represent and explain numerical and algebraic relationships.
  
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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