Title of Activity s2

Surface Area Wrap

Author(s)

Sarah Pumphrey and Gabrea Bender

Subject(s)

Geometry

Grade Level

Duration

Five 45 to 50-minute periods

Rationale (How this relates to engineering)

Surface area is a crucial concept in many engineering applications. In construction, engineers are responsible for providing adequate paving material to cover surfaces without wasting resources by having excess. The surface area of a cross section of a beam helps a structural engineer prove that the beam he or she has selected will safely support a building. Hydraulic engineers model the flow of various liquids by utilizing the surface area measurement of a cross section of pipe.

Activity Summary

In this lesson activity, students explore relationships between the surface area of individual polygons and a three dimensional object. First, students practice the fundamentals of calculating surface area on various polygons, three dimensional objects, and circles. Then they work with a partner to record the dimensions of an unusually shaped box, calculate the surface area, and cut paper to the ideal size to wrap the box. Finally, they calculate the maximum efficiency of a real-world surface area problem by applying the concept of using only what is necessary.

Objectives

Upon completion of this lesson, students will be able to…

1. Calculate the surface area of any polygon

2. Classify three-dimensional geometric figures

3. Apply their knowledge of two-dimensional surface area calculations to a three-dimensional object

4. Explain properties of real life surface area problems

5. Justify the expense related to material usage based on mathematical reasoning

Standards

Mathematics

· MA-HS-3.1.9 – Students will classify and apply properties of three-dimensional geometric figures.

· MA-HS-3.1.11 – Students will visualize solids and surfaces in three-dimensional space when given two-dimensional representations (e.g., nets, multiple views) and create two-dimensional representations for the surfaces of three-dimensional objects.

· MA-11-2.1.1 – DOK 3 – Students will determine the surface area and volume of right rectangular prisms, pyramids, cylinders, cones, and spheres in real-world and mathematical problems.

· MA-11-2.2.1 – Students will continue to apply to both real world and mathematical problems using U.S. customary and metric systems of measurement.

Background Knowledge

Students should have previous knowledge of surface area calculations for simple shapes. They should recognize that the surface area of any polygon can be found by calculating the sum of the surface areas from familiar shapes within the complex polygon.

Basic Shape Surface Area Equations:

Square:

SAsquare = b2 = base squared

*This shows the basic surface area concept, thus the units are “squared”

Rectangle:

SArectangle = b*h = base * height .

*Expand the previous concept to any rectangle

Parallelogram:

SAparalleogram = b*h = base * height

*A perpendicular line from one of the obtuse angles to the base creates a triangle which could slide to the opposite side and create a rectangle with the same area as the parallelogram.

Triangle:

SAtriangle = ½ b*h = ½ base * height

*Dropping a line from the top vertex perpendicular to the base creates a right triangle. Any right triangle can be doubled and flipped to produce a rectangle that is twice the surface area of the original right triangle. Thus the triangle surface area equation is one half of the rectangle surface area equation.

Trapezoid:

SAtrapezoid = ½ h (b1 + b2) = ½ height * (first base + second base)

*A rectangle with the same height as this trapezoid, but with a base the sum of the top and bottom would have an area twice the size of the trapezoid.

Any Polygon:

Divide any polygon into segments which have regular shapes. Calculate the surface area of those individual shapes, and add them together to find the surface area of the entire polygon.

Circle:

SAcircle = pr2 = 3.14 * radius squared

Solid Object Surface Area and Volume Equations

*The surface area for any flat-plane, geometric object can be calculated by summing the surface area of the shapes which form the outer surfaces of the object.

Rectangular Prism:

SAbox = 2(bh + bw + hw)

= 2 * [(base * height) + (base * width) + (height * width)]

*Three different squares are doubled: the front face, the side face, and the top face.

Volumebox = bhw = base * height * width

Triangular Prism:

SAwedge = bh + w(b + a + c)

= (base * height) + width *(base + side a + side c)

*Doubling the triangular face (½ bh) gives a parallelogram with the surface area of base times height. Then add the surface areas of three rectangles, the bottom, the side and the other face.

Volumewedge = ½ bhw = ½ * base * height * width

Cylinder:

SAcylinder = 2pr2 + 2prh = 2 *circular ends + circumference * height

Volumecylinder = pr2h = area of base * height

Sphere:

SAsphere = 4 pr2

Volumesphere = 4/3 pr3

Pyramid:

SApyramid = Sum of (Area of Faces) + Area of Base

Volumepyramid = 1/3 Bh = 1/3 * area of base * height

Cone:

SAcone = pr2 + prl

Volumecone = 1/3 pr2h = 1/3 * area of base * height

Additional Resources

Animated Polygons: http://plus.maths.org/issue27/features/mathart/index.html

Unfolding polyhedra: http://plus.maths.org/issue27/features/mathart/index.html

Think Quest Geometry: http://library.thinkquest.org/C0110248/geometry/menareapyramid.htm?tqskip1=1

Materials Required

Containers with geometrically flat sides (1 per student)

Scissors

Wrapping Paper

Worksheets

Pencils

Tape

Graph paper

Glue / Glue Stick

String

Rulers

Activities

Part 1 – (allow approximately 45 to 50 minutes for each part)

Begin with a review the names of all basic geometric shapes and their definitions (circle, square, triangle, rectangle, trapezoid, rhombus, pentagon, hexagon, octagon, nonagon, etc.). The students will generate a list of surface area equations for as many of the shapes as they already know, and will then receive a hand out with images and equations. The instructor will discuss theories of the surface area equations and show an interactive web site “Animated Polygons” as a visual aid. The class will then calculate the surface area of commonly found objects using the in-class Smartboard. Students work will be saved and given to them the following day as a review sheet.

Part 2 –

Part 3 –

Homework: Students will calculate the surface area of an object from their home.

Part 4 –

The students will work individually or in pairs to complete a hands-on, authentic learning activity. They will follow instructions in a work packet, which will instruct them to produce drawings of a three dimensional object, label their drawings, calculate the surface area of their object, and cover the object with wrapping paper.

Part 5 –

Extending the Lesson – Cylindrical, star, and other various shaped boxes (which can be purchased at local craft or hobby stores, or using a website such as Paper Models of Polyhedra, http://www.korthalsaltes.com/index_hoofd2.html for paper models of these shapes) make the lesson more challenging for advanced students. Advanced math students will also benefit from the challenge of the “advanced” assessment because links the math to a real-world application.

Assessment of Student Learning

1. Student Participation

2. Living with Polyhedrals (Homework)

3. Surface Area Wrap Packet (Group Work In Class)

4. Solving for Surface Area (Individual Assessment)

Assessment of the Activity

[Type Here]

Student Quotes

“I don’t normally want to go up [in front of the class], but I just want to write on the Smartboard.”

Reflections

As many beginning teachers discover, lesson plans have to be adaptable to the individual group at a certain time. Originally, we had planned for this lesson to be two days worth of material. It has since been expanded into four (one 90-minute block, and three 50-minute class periods). This lesson plan has been re-written to reflect what was taught, however, for teacher scheduling purposes, it was written as though there were five equal classroom periods. As we taught, we incorporated more hands-on activities, such as the 3-D paper models, and decided to give the homework after completing the packet, prior to the quiz.