Surface Area and Volume

Abstract

This lesson is designed to introduce students to surface area and volume.

Objectives

Upon completion of this lesson, students will:

  • have been introduced to the notion of surface area and volume.
  • have learned the terminology used with surface area and volume.
  • have experimented with the surface area and volume of different prisms.

Standards Addressed

Grade 6

  • Geometry

    • The student solves problems (including real-world situations) using perimeter, area, or volume.

Grade 7

  • Geometry

    • The student solves problems (including real-world situations).

Grade 8

  • Geometry

    • The student solves problems (including real-world situations).

Grade 9

  • Geometry

    • The student solves problems (including real-world situations).

Grade 10

  • Geometry

    • The student solves problems (including real-world situations).

Third Grade

  • Measurement and Data

    • Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
    • Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

Fifth Grade

  • Measurement and Data

    • Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

Sixth Grade

  • Geometry

    • Solve real-world and mathematical problems involving area, surface area, and volume.

Seventh Grade

  • Geometry

    • Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Geometry

  • Geometric Measurement and Dimension

    • Explain volume formulas and use them to solve problems
    • Visualize relationships between two-dimensional and three- dimensional objects

Grades 3-5

  • Measurement

    • Apply appropriate techniques, tools, and formulas to determine measurements

Grades 6-8

  • Geometry

    • Use visualization, spatial reasoning, and geometric modeling to solve problems

  • Measurement

    • Apply appropriate techniques, tools, and formulas to determine measurements

Grades 9-12

  • Geometry

    • Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships

  • Measurement

    • Apply appropriate techniques, tools, and formulas to determine measurements

Grade 7

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 2: The learner will understand and use measurement involving two- and three-dimensional figures.

Geometry

  • Geometry and Measurement

    • Competency Goal 2: The learner will use geometric and algebraic properties of figures to solve problems and write proofs.

Technical Mathematics I

  • Geometry and Measurement

    • Competency Goal 2: The learner will measure and apply geometric concepts to solve problems.

Technical Mathematics II

  • Geometry and Measurement

    • Competency Goal 1: The learner will use properties of geometric figures to solve problems.

Integrated Mathematics III

  • Geometry and Measurement

    • Competency Goal 2: The learner will use properties of geometric figures to solve problems.

6th Grade

  • Measurement

    • The student will demonstrate through the mathematical processes an understanding of surface area; the perimeter and area of irregular shapes; the relationships among the circumference, diameter, and radius of a circle; the use of proportions to determine
    • The student will demonstrate through the mathematical processes an understanding of surface area; the perimeter and area of irregular shapes; the relationships among the circumference, diameter, and radius of a circle; the use of proportions to determine unit rates; and the use of scale to determine distance.

5th Grade

  • Measurement

    • The student will demonstrate through the mathematical processes an understanding of the units and systems of measurement and the application of tools and formulas to determine measurements.

7th Grade

  • Measurement

    • The student will demonstrate through the mathematical processes an understanding of how to use ratio and proportion to solve problems involving scale factors and rates and how to use one-step unit analysis to convert between and within the U.S. Customary
    • The student will demonstrate through the mathematical processes an understanding of how to use ratio and proportion to solve problems involving scale factors and rates and how to use one-step unit analysis to convert between and within the U.S. Customary System and the metric system.

8th Grade

  • Measurement

    • The student will demonstrate through the mathematical processes an understanding of the proportionality of similar figures; the necessary levels of accuracy and precision in measurement; the use of formulas to determine circumference, perimeter, area, and
    • The student will demonstrate through the mathematical processes an understanding of the proportionality of similar figures; the necessary levels of accuracy and precision in measurement; the use of formulas to determine circumference, perimeter, area, and volume; and the use of conversions within and between the U.S. Customary System and the metric system.

7th Grade

  • Measurement

    • 7.8 The student will investigate and solve problems involving the volume and surface area of rectangular prisms and cylinders, using concrete materials and practical situations to develop formulas.

8th Grade

  • Measurement

    • 8.7 The student will investigate and solve practical problems involving volume and surface area of rectangular solids (prisms), cylinders, cones, and pyramids.

Secondary

  • Algebra II

    • AII.16 The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve practical problems, including writing the first n terms, finding the nth term, and evaluating summation formulas. Notation will include Σ and an.
    • AII.16

Textbooks Aligned

6th

  • Module 7 - Wonders of the World

    • Section 1: Three-Dimensional Geometry
      • Reason for Alignment: Surface Area and Volume lesson guides the students through relationships in geometry. The textbook does not address surface area as such at this point, but volume and area are explored, so this shouldn't be a problem if the teacher wants to take the class a little further with the worksheet used in the lesson.

7th

  • Module 6 - Flights of Fancy

    • Section 3: Square Roots, Surface Area, and Area of a Circle
      • Reason for Alignment: This lesson can be used for the section on surface area at this time, if that fits the needs of the teacher and the class. The discussion and the worksheets can easily be adapted.

  • Module 7 - MATH-Thematical Mix

    • Section 1: Measurement and Volume
      • Reason for Alignment: This lesson was recommended in section 1 for its use of surface area. It can be used effectively here now as well, especially because students are now working with volume too.

8th

  • Module 4 - Inventions

    • Section 1: Circumference, Area, and Volume
      • Reason for Alignment: This lesson only uses for prisms so could be applied for that part of the section only. The accompanying worksheet contains questions for use with rectangular and triangular prisms.

  • Module 5 - Architects and Engineers

    • Section 4: Surface Area and Volume
      • Reason for Alignment: Though this lesson has been recommended in other modules it would fit here as well. It accompanies the Surface Area and Volume activity.

Student Prerequisites

  • Arithmetic: Students must be able to:
    • perform integer and rational arithmetic
  • Technological: Students must be able to:
    • perform basic mouse manipulations such as point, click and drag
    • use a browser for experimenting with the activities

Teacher Preparation

Key Terms

surface area

A measure of the number of square units needed to cover the outside of a figure

volume

A measure of the number of cubic units needed to fill the space inside an object

Lesson Outline

  1. Focus and Review

    Remind students of what they learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson:

    • Ask students if they recall how to find the area of a two-dimensional shape, such as a square.
    • Entertain a discussion on surface area and volume of three-dimensional figures.

  2. Objectives

    Let the students know what they will be doing and learning today. Say something like this:

    • Today, class, we will be talking about the surface area and volume of three dimensional figures.
    • We are going to use the computers to learn about surface area and volume, but please do not turn your computers on or go to this page until I ask you to. I want to show you a little about the program first.

  3. Teacher Input

    • Open your browser to Surface Area and Volume in order to demonstrate this activity to the students.
    • Show the students how to choose between a triangluar prism and a rectangular prism. You may need to briefly discuss what these figures are.
    • Show students how to use the scroll bars to adjust the length, width, and height of the prisms.
    • Show students how to adjust the view of the figure using the three buttons at the top of the applet. Talk about what each view is showing. Be sure to show the students that clicking on the figure and rotating it will allow them to view it from any angle they choose.
    • Point out to students the place where the surface area and volume of the figures is displayed.

  4. Guided Practice

    • Once you have verified that there are no questions regarding the use of the applet, pass out the Worksheet To Accompany "Surface Area and Volume"
    • Walk the students together through question 1 on the worksheet -- make sure they understand the slider bars for controlling size.

  5. Independent Practice

    • Allow the students to work independently or in small groups on the rest of the worksheet as you monitor the room for progress.
    • Have the students repeat the worksheet for the triangular prism, commenting on how finding each answer on the sheet changes from the rectangle to the triangles.

  6. Closure

    • You may wish to bring the class back together for a discussion of the findings. Once the students have been allowed to share what they found, summarize the results of the lesson. It is important that every student know the correct answers to the worksheet in order to fully understand surface area and volume.

Alternate Outline

This lesson can be rearranged for the one computer classroom. Have the students work through the worksheet by directing your movements on the computer.