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VOLUME AND SURFACE AREA: Which is More? (Grades 6-8)

MASTER TEACHER

Yvette Barnes

OVERVIEW

This lesson is designed to help students discover and calculate the volume and surface area of three dimensional figures. Through the use of video, students become aware of different three dimensional shapes, their uses in real-life applications, and methods used to calculate their volume and surface area. Hands on activities and student interaction are incorporated to assist with exploring, discovering, and calculating the volume and surface area of different shapes.

ITV SERIES

Math Vantage, Lesson 11: Containers: Surface Area and Volume

LEARNING OBJECTIVES

Students will be able to

measure volume by liquid displacement
measure surface area using grid paper
derive formulas from experimentation and reasoning
apply formulas to compute the volume of rectangular prisms, cylinders, pyramids, cones, and spheres
apply the knowledge that either surface area or volume can remain constant while the other changes.

SC MATH/SCIENCE STANDARDS MET: MATHEMATICS

Numerical and Algebraic Concepts and Operations -Middle Grades Represent situations and number patterns with models, tables, graphs, verbal rules, and equations and make connections among these representations.

Geometry and Spatial Sense - Middle Grades Model, identify, describe, and compare two- and three- dimensional geometric figures.

Use technology whenever appropriate to explore concepts and applications of geometry

Develop spatial sense by thinking about, constructing, and drawing two and three dimensional geometric figures.

Investigate and predict the results of combining, partitioning, and changing shapes, figures, and models.

Measurement Middle Grades
Extend their understanding of the concepts of length, capacity, weight (mass), perimeter, area, and volume.

Use concrete and graphic models to discover the formulas for finding perimeter, area, and volume of common two- and three- dimensional shapes.

Use measurements and formulas to solve real world and mathematical problems.

SCIENCE:
Area III: Matter and Energy Strand 3: Forces produce physical and chemical changes and, in these changes, matter and energy are conserved.

describe and measure qualities such as time, distance, displacement, mass, force, friction, velocity, acceleration, momentum, potential energy, and Kinetic energy that characterize changes in position or motion of objects.
describe and measure melting point, boiling point, mass, volume, density, concentration, solubility, and acidity/ basicity that are subject to change in chemical systems.

MATERIALS

containers of various shapes and sizes
three-dimensional solids
three-dimensional paper models for cut out (1 for each student)

containers that have the same volume but different shapes
100 ml graduated cylinders (1 for each group of 4 students)
500 ml beakers (1 for each group of 4 students)
marbles (1 marble for each group of 4)
cubes (27 for each group of 4) water
hard-boiled eggs (1 for each group of 4)
scissors
centimeter grid paper (20 sheets for each group of 4)
transparent tape

PRE-VIEWING ACTIVITIES
Say, "Today we are going to discover and calculate the volume and surface area of various shapes.

The teacher will display boxes and plastic containers in various shapes and sizes. The teacher will ask the students to examine the containers and make predictions about which container has the greatest capacity.

FOCUS FOR VIEWING
Ask the students to watch the video for examples of three-dimensional figures, methods used to compute the volume and surface area, and how volume and surface area dictate the shape and size of plastic containers.

VIEWING ACTIVITIES
Start the video at the beginning of Lesson 11: Containers: Surface Area and Volume

Pause the video after she says, "Containers come in all shapes and sizes." Ask the students to list examples of three-dimensional shapes. Resume.

Pause the video after she says, "rectangular prism cooler."

Discuss the shapes (cone, sphere, cylindrical, frustum, ellipsoid, parabloid, torus, frusta, and rectangular prism) that were found in the ice cream shop. Resume.
Pause the video after the sphere burst. Ask the question, "What's important about a sphere?" (Answer: The sphere gives the most volume for most surface area.) Resume.

Pause the video after she says, "Why are there so many containers and why are they the shapes that they are?" Ask the students to give their opinions. Resume.

Pause the video after Roger says, "recycle." Ask the students to answer the question, what does Roger consider when deciding on a particular container? (Answer: Volume, surface area, strength, stackability and customer expectations dictate the shape and size of containers.) Resume.

Pause the video after "measure volume in quarts, gallons, liters, and milliliters." Ask students to list the methods that were used to find the volume. Resume.

Pause the video when the basketball appears. Discuss different shapes and the surface area of these shapes. Resume.

Pause the video after she says, "Don't lose your head." Discuss the method used to find odd shaped things. Resume.

Stop the tape when she says, "Which pop can do you think will float?" Discuss her method for finding the volume using displacement.

SUMMARY OF VIDEO

Ice cream, candy, and swimming pools are used to illustrate the variety of shapes and names for three-dimensional figures. A manufacturer of plastic containers comments that volume, surface area, strength, stackability, and customer expectations dictate the shape and size of containers. Methods to find surface area and volume are illustrated visually with the examination of patterns and the stacking of cubes. Volume computation by displacement of water is also presented.

POST VIEWING ACTIVITIES

Using the Activity Sheets, each student will construct a rectangular prism, cube, cylinder, pyramid, and cone from grid paper models. The students will be able to use these as models of three-dimensional shapes. Students will use cubes and grid paper to explore the volume of rectangular prisms and cubes. The students are asked to work in groups of four. Each group is supplied with a stack of centimeter grid paper. The teacher should demonstrate cutting one square out of each comer of the paper, folding one row and one column up and taping them to form a box (rectangular prism or cube) with no lid. Each student makes a similar box. The teacher asks if there are any other boxes that can be made. Each group is to work on making boxes of different sizes to find out if all the boxes have the same volume and surface area. If not, which box has the greatest volume? Greatest surface area? A discussion about how this information is used in packaging and marketing products, such as cereal and cookies follows. Different boxes are shown as examples. The length, height, and volume are recorded in a table by the teacher. Students should be able to develop a formula for computing the volume and surface area of a rectangular prism and a cube.

Students will use the displacement method to find the volume of a marble and a hard-boiled egg. The students are asked to work in groups of four. Each group is supplied with a 500 milliliter beaker and a 100 milliliter graduated cylinder. The students will partially fill the cylinder with water. The students will record the volume of the water. Carefully drop the marble into the cylinder. Record the volume reading now. Determine the volume of the marble by subtracting the initial volume reading from the reading after placing the marble in the cylinder. Repeat this method using the egg and the beaker.

ACTION PLAN

Ask each student to bring his/her favorite candy bar. Each student is to measure the length, width, and height of his/her candy bar to the nearest centimeter. Each student is to determine the volume of his/her bar by multiplying length times width times height. Also each student is to find the surface area of his/her bar. The results will be recorded on the board. Students will make comparisons and determine which candy bar has the largest volume and which has the largest surface area. Students will be asked to research the origins of their candy bars. Students will be encouraged to write the companies that manufactured their candy bars to obtain information.

EXTENSIONS

Using the candy, have the class write and prepare a TV commercial that would persuade people to want to buy your candy bar.

Investigate how surface area and volume are related. Students will work in groups of three. Find the height and radius of the base of a soft drink can to the nearest millimeter. Find the volume and surface area. Record your results. Alter the height and radius measurements to create a new cylinder that has approximately the same volume as the soft drink can. Then find the surface area of this cylinder. Record your findings. Use your calculator to create four more cylinders that have approximately the same volume as the soft drink can. Record your findings. What dimensions resulted in the greatest surface area? What dimensions resulted in the least surface area? When the radius is greater than the radius of the original can, what happens to the height of the cylinder?

Investigate density and buoyancy.