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Insides and Outsides

Summary

This activity will teach the basics of surface area and volume.


Materials

Invitation to Learn

  • Cubes
  • Chart Paper

The Great Cover Up

  • X/Y-axis Dry Erase Mats
  • Key Graphing Cling
  • Pre-made rectangular prisms
  • Rectangular prism net patterns
  • Markers
  • Cubes
  • The Great Cover Up (pdf)
  • Graph paper

Cut it Out!

Additional Resources

Books

Math Dictionary-The Easy, Simple, Fun Guide to Help Math Phobics Become Math Lovers, by Eula Ewing Monroe; ISBN 978-1-59078-413-6


Background for Teachers

In order for students to visualize and determine surface area and volume of three-dimensional shapes, it is important to manipulate actual objects. They must also have experience with the concepts of area, nets and rectangular prisms prior to the introduction of this standard and objective (Standard IV Objective 2). When talking about area, we are referring the measurement of a two-dimensional shape. When talking about surface area, we are referring to the measurement of a three-dimensional shape.

The surface area of a prism is the sum of the areas of all the faces, including the bases. The surface area is measured in square units. Although the students may not be familiar with this concept, they have actually experienced it as they've worked with nets. Surface area takes nets one step further by determining actual measurements. The purpose of this lesson is to help the students make this connection. The mathematical formula for surface area is:

SA= 2(l · w) +2(l · h) + 2(w · h)

The actual formula for surface area involves using length, width and height, at this level. Having the students find the area of each face and then adding them together as square units to find the total will be less confusing for fifth graders.

The volume of a prism tells how many cubic units it takes to fill the prism. Volume is measured in cubic units. The mathematical formula for volume is:

V= l · w· h


Instructional Procedures

Invitation to Learn

  • Put students into small groups of two to four, and give each group a container with at least 48 cubes.
  • Challenge each group to build as many different regular rectangular prisms that have a volume of 12 cubic units.
  • Once a prism has been built, ask them to set it aside to keep as an example so that the same prisms are not repeated. (With 12 cubes, they can build four different rectangular prisms: 1x2x6, 1x3x4, 1x1x12, and 2x2x3.)
  • Demonstrate the four different prisms that twelve cubes can make by having a few volunteers stand and describe their prisms -- length, height, and width. Because of previous lessons on volume, the students should be able to describe their prisms using length, width and height. A discussion on whether a different orientation makes a difference in the dimensions may be needed.
  • Review the definition of volume with your students.
  • As a class, find the volume for the four different prisms using length, width, and height.
    Length Width Height Volume
     
     
     
     
     
     
     
     
     
     
     
     
     
  • On chart paper, record the students findings . Highlight the fact that because each prism is made of twelve cubes, each has a volume of 12 cubic units.

Instructional Procedures

The Great Cover Up

Use the concept of area with 2-D measurement, using a 3-D prism (1 x 2 x 3), to introduce the concept of surface area or 3-D measurement. Ask the students to predict if the prisms have the same volume, do they have the same surface area. Come back to this question at the end of the Cut it Out! activity.

  • Show the class the pre-made prism or 3-D object and ask the students how they can use area to determine how much wrapping paper would be needed to cover the entire prism without any overlapping.
    -- This measurement is called surface area and can only be found for 3-D objects.
  • Discuss their ideas for ways to measure the surface area.
  • If they struggle coming up with a solution, bring out nets that were used in previous lessons, to visualize the connection between nets and surface area.
  • Review area and its formula by finding the area of each face. This discussion is critical in helping students make the connection between area and surface area.
  1. With the help of the students, use the Key Graphing Cling to model how to draw the net of the pre-made prism.
    • Use the concept of a room to help them visualize each surface as it is being drawn: floor, ceiling, four walls: two front/back walls, two side walls.
    • Use these same terms to label each part of the net.
    • Discuss how to find the area of each face, leading to the idea that adding all areas would give the total area or surface area.
    • Discuss how using just letters rather than whole words as labels can be simpler.
    • Have students come up with a formula for surface area using only letters:
      SA = a + a + b + b + c + c
      (f=area of floor and ceiling; b=area of front and back wall; c=area of side walls)
    • Plug in actual area for each face underneath formula. Give students The Great Cover Up to record data with you.
    • Repeat this process of creating nets together using different rectangular prisms.
  2. Next, pass out individual 9x11 Double sided X/Y-Axis Dry Erase Mats and dry erase markers so students can draw more nets with you. Continue to use worksheet to determine and record the surface areas.
  3. Now using graph paper and the cubes.
    • Have students work in small groups to reconstruct the original four 12-cube prisms and have them draw nets for each one.
    • Fill out The Great Cover Up for each prism.
    • Save their nets for the next activity.

Cut it Out!

  1. Use chart made in Invitation to Learn and referring to the first prism listed, have all students find that net and cut it out.
  2. Fill out #1 on Cut it Out together.
  3. Students will repeat the process for the remaining three prisms.
  4. Review their predictions of the connection between surface area and volume.
  5. In their math journals, have students write what they learned about surface area and volume. Have them analyze the pattern they learned from their Cut it Out. Have the students explain how to use a formula to find volume, and how they would find the surface area of a prism.


Extensions

Attachments

  • Prism Race (pdf)
  • Challenge students to find out how many different rectangular prisms they can make with a volume of 36 cubes. Encourage them to think of ways to make sure they have made all possible prisms.
  • Challenge students to make as many prisms as possible with the same surface area. Have them record the volumes of their solids and note any patterns.
  • Art: Draw three-dimensional rectangular prisms on isometric dot paper.
  • Design a net that can be folded into a rectangular prism that can hold 24 Multilink cubes.

Family Connections

  • Have students find at least five rectangular prisms from household items. Have them measure the length, height, and width of each item. Record the measurements in their math journals, and then find the volume and surface area of each of the items.


Assessment Plan

Attachments

  • Informal assessment includes class discussion, math journals and observation of group/partner work.
  • The Great Cover Up
  • Cut it Out
  • Birthday Boxes


Bibliography

Sowel, E.J. (1989). Effects of manipulative materials in mathematics instruction. Journal for research in mathematics education, 20 (4), 498-505.

This review of research sums up the result of sixty studies addressing the effectiveness of manipulatives on student learning and attitudes in mathematics teaching. Sowell concludes that the more concrete the manipulatives, and the longer the time spent using them, the better instructional outcomes.

Hinzman, K.P. (1997). Use of manipulatives in mathematics at the middle school level and their effects on students' grades and attitudes. ERIC Source (ERIC # ED411150). Retrieved December 10, 2006, from http://www.eric.ed.gov

This paper reports on a study that examines mathematical scores when hands on manipulatives and group activities were used in the classroom. Results indicate that student performance was enhanced by the use of manipulative materials; and students' attitudes toward mathematics were significantly more positive than those in previous years when manipulatives were not used.


Created: 07/16/2007
Updated: 01/31/2018
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