Investigating Surface Area

This is a hands on lesson best used to introduce geometry students to 3-dimensional figures. Students will have the opportunity to draw 3-dimensionally and create collapsible figures which can be used to develop the standard surface area formulas.

A lesson plan for grades 6–7 Mathematics

By Jennifer Bronzini

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Related topics

• Learn more about mathematics.

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Learning outcomes

Students will:

• create, draw and fold 3-dimensional figures.
• make 2-dimensional nets for 3-dimensional solids.

Teacher planning

Time required for lesson

90 Minutes

Materials/resources

• Graph paper (larger grid sizes are better)
• Rectangular dot paper
• Isometric dot paper (dots are arranged in triangles)
• Scissors
• Tape
• Rulers

Technology resources

None.

Pre-activities

• Ask students to name 3-dimensional figures for which they know the correct geometric term (correct terms as necessary--for example: if a student says “box”, let them know that is a “rectangular prism.”
• Ask students to name real objects that exhibit the forms they name (buildings, natural phenomena, etc.)

Activities

Introduction:

1. Let students know that the topics for the next unit are moving from 2-dimensional to 3-dimensional.
2. Today will lay the basis for some of the formulas they have learned in the past.

Body:

1. Surface Area
• Definition: the sum of the area of the faces of a 3-dimensional object
• Ultimately want to develop formulas for the surface area of polyhedra (solids with flat faces).
2. Two Types of Solids
• Prism
• Two parallel faces called BASES
• Other bases are parallelograms (LATERAL FACES)
• Pyramid
• All faces but one intersect at a common point
• VERTEX--common intersection point of faces
• BASE--face that does not intersect at common point
• All lateral faces are triangles
3. Special Types of Graph Paper
• ISOMETRIC DOT PAPER--graph paper with dots arranged in triangles; makes 3-dimensional drawing easier.
• RECTANGULAR DOT PAPER--graph paper with dots arranged in squares; allows drawing of “unfolded” solids called NETS.
4. Steps to draw figures 3-dimensionally
• Use isometric dot paper.
• Draw the top of the “box” first.
• Use horizontal connections to indicate one length of “box”.
• Use diagonal connections to indicate “box” width.
• Draw 2nd length by connecting ends of diagonal width lines.
• Draw the height of the box by using vertical dots to draw straight lines of desired length from each of the four corners of the box top (NOTE: on isometric dot paper the vertical space between dots is 2 units long.).
• Draw the bottom of the box by connecting the bottoms of the heights drawn in step 3.
• Hidden edges need to be shown with dotted lines. Edges to alter are: bottom back, bottom left, and back left height edge.
5. Guided Practice
   Have students draw a rectangular prism 5 units high, 6 units long and 3 units wide.
6. Steps for drawing nets
   (**A NET IS AN “UNFOLDED” SOLID**)
   (**THIS IS A VERY VISUAL PROCESS AND GIVES MANY HIGH SCHOOL STUDENTS PROBLEMS**)
• Use rectangular dot paper or graph paper.
• In the center of the paper draw the bottom of the “box” - the dimensions are the length and width given. Label “BOTTOM”.
• Above and below the “BOTTOM” attach the lateral sides connected to it — these are the front and back of the box visually. The dimensions are the length and height given. Label these “FRONT” and “BACK”.
• Below the “BACK” piece attach the top of the “box”. The dimensions are the same as the bottom (length and width given).
• To the right and left of the “BOTTOM” attach the ends of the “box”. The dimensions are the width and height given. Label these each “END”.
7. Guided Practice
   Have students draw the net for the rectangular prism they have drawn. After the net is drawn have students cut along the perimeter of the diagram and fold the prism to create a 3-dimensional model of the figure.
8. Independent Practice
• Provide students with isometric and rectangular dot paper.
• Give them pictures of several rectangular solids and pyramids. The diagrams need to have dimensions labeled.
• Have students create their own 3-dimensional drawings and nets for the figures.

Assessment

• Ask students to use a net from one of their rectangular solids to develop the formulas for the lateral and total (surface) area of the solid.
• Ask students to show what happens to the surface area of a rectangular prism if the length and height are doubled.

Supplemental information

None.

Related websites

Space Figures and Basic Solids
http://www.mathleague.com/help/geometry/3space.htm

A Real World Geometry Project
http://www.wested.org/pblnet/exp_projects/realworld.html

Comments

Regular level geometry students enjoy this hands on opportunity very much. So many of their difficulties with 3-dimensional concepts are visual. These activities help them to break down the figures into pieces they can see. Be wary of lots of little pieces of paper on the floor after you are done!!