Grade:  6, 7, 8

Objectives:
    1. Perimeter, area, circumference, and volume of plane and solid geometric figures will be reviewed using measurements, diagrams, and/ or calculations.

    2. Students will work in small groups to explore hands-on applications of square and cubic units, develop concepts of surface area, work with and calculate volume, and explore making various polyhedra.
 

Time required: 5 - 8 days

Instructional Procedures:
         Groups:
        These lessons may be done in random order thus facilitating use of computers and group rotations.  It is suggested that Lessons 1 and 2 be done with teacher facilitation prior to the subsequent lessons.  One computer per group is highly encouraged, but not manditory for use of these lessons.

         Depending on access to computers:
         A.  If one computer is available, teacher may connect to a large screen tv for whole class viewing of the lesson or may send small groups to the single computer for viewing.
        B.  If no computer is available, teacher may wish to print directions and worksheets for lessons and place them in a folder for each group.
       C.  If multiple computers are available, each group of 3 to 5 students should be assigned a computer.

        Whole Class:

Lesson 1:
     A.  Review formulas for calculating area of various surfaces.  Access Dave's Math Tables  with class as a whole having students cite instances for using area of various shapes. (Sow grass seed, paper or paint a wall, buy carpet, layout a baseball diamond, or install an oval pool).
     B.  Teacher will review formulas and calculating area Area and Volume with class as a whole.

Lesson 2:
    A member of each group should access Volume and Surface Area and will explain this activity to members of their group. Each group will glue sugar cubes together to form teacher designated prisms, i.e. 4 x 6 x 3, 5 x 7 x 4, etc.. Student groups will paint the exterior surfaces of the prisms, then detatch the individual cubes and observe the number of faces painted. The teacher will facilitate each group (by asking questions) to develop a formula for finding surface area based on the painted surfaces of their prisms.
   Students will work in small groups to disassemble a box and calculate the areas of the individual parts.  Through their work together, they will confirm the formula for calculating total surface area of a prism.. They will discuss as a class or small group how that formula would change if the base had a different shape.  Students will complete the Find Surface Area of a Prism  and turn it in for grading.

Lesson 3:
    A second student from each group will access  Total Surface Area: Learning By Logic, will facilitate work in their groups with geoboards (Geometer's Sketchpad or graph paper) to calculate area of plane surfaces, and will then use their newly developed formula in calculating total surface areas of five or more rectangular prisms in the classroom.  Student groups will hand in a list of prisms measured, formulas used, calculations, and total surface areas of the prisms.  Students may want to have a peer check their work before handing in their list (such as below) for evaluation.
 

Lesson 4:
    Using the formula for surface area, each student will measure his/ her math book, calculate the amount of contac paper needed (allowing 2 inches overlap on each edge, and will make a book cover (first from newspaper, then after correcting any errors, from contac paper).  Again, peer checking for a snug fit and the two inch overlap may be used to help provide accuracy.

Lesson 5:
    Students will use geoboards to illustrate a triangle. Following their previous work, student groups will derive a formula for finding the surface area of a triangular prism. When each group has arrived at a consensus, their formula will be written on the board. Group spokespersons may be called upon during discussion to explain their formulas. Discussion should produce a workable formula for finding surface area of a triangular prism. An attached Worksheet #2 may be copied for individual practice and may be graded as an assessment

Lesson 6:
    Student groups will access Making Polyhedra Activity and follow directions in the Activity to construct prisms and polyhedra. (Teacher needs to make copies ahead of time.) {Straws and string, or gumdrops and toothpicks may be used to enable students' visual perception and hands-on exploration of various polyhedra.  Students should use these manipulatives to make a chart of polygons, their numbers of faces, edges, and vertices - Euler's Rule} Worksheet: Polyhedra Shapes.

Lesson 8:
    The class will use Filling and Wrapping and will work in groups to determine what box size is needed for a particular size ball, draw the shipping box, and build the box. Group discussions will include finding the surface area and volume of the object to be packaged in order to make a snug package casing for the object. (This activity could be related to Careers by investigating (on-line) job/career opportunities in packaging. How much money is spent annually on packaging? How many people are hired by local department stores to wrap packages especially during the Christmas holidays? Interview someone.)  The accompanying Shipping Box Worksheetshould be completed.  A Rubric for evaluating this activity is listed.

Lesson 9:
    Student groups will review Volume of a 12 Ounce Can and will discuss within their group how to find the volume and surface area of a cylinder.
     Students will work with construction paper to produce a cylinder. (Several cyclindrical cans should be available for examination including one or more empty Pringle chip containers). When consensus as to  formulas for finding the volume and surface area of a cylinder within a group is reached,  those formulas will be posted on the chalkboard. When all are finished, whole class consensus should be sought. (If necessary, a Pringle can should be sliced and the resulting net should be sketched.)
    The Worksheet #5 should be completed by each student and results checked together as a whole class with explanations for any questions coming from students whenever possible. Quiz

Lesson 10:
    Students will summarize their activities with Surface Area and Volume, will select an irregular shaped object to wrap and will draw the net, calculate using correct formulas, and explain in writing how to determine the amount of wrapping paper required for this object.  Completed designs (nets) and summaries may be shared with the class.  Summative Assessment.
 

Assessment:
        Attached are:
                    Quiz on Surface Area and Volume of Cylinders.
                    Rubric for Shipping Box Activity
                    Summative Assessment for Surface Area and Volume of Cylinders and Prisms
*** All worksheets, constructions, and written summaries/ explanations will be collected in a portfolio for assessment.

Materials Needed:
    sugar cubes
    paint, brushes
    geoboards
    dot paper, graph paper, drawing paper, construction paper, wrapping paper
    rulers, meter sticks, tape measures
    newspapers  (table coverings)
    cylinders including pringle cans
    scissors
    glue
    empty boxes - cereal boxes, shoe boxes, etc that students can easily disassemble
    cardboard and something to cut cardboard
    flexible straws, scotch tape
    paper nets
    contact paper
 
 

    Name _______________                                      Date _____________

Each group will need a small rectangular box they can open into a flat surface.

1. Remove the tape or other material holding the box together and spread out the parts on a flat surface.

2. Each student should sketch the "net" of the box.

3. Work together in your group to measure the length of each side of each section of the open box.
    bottom = _______ in. x ____________ in.

4. Measure one of the ends of the box: ___________ in x __________ in.

5. Now measure the front of the box:  ____________in x __________ in.

6. With a pen or marker, label the measurements on the box.

7. Are some of the measurements the same?  ____  Which ones? _________

8. Are there sections of the box that have exactly the same measurements? ________  Would this be true
for any rectangular prism? _____

9. Work with your group to find the area of each section of the box, add them together and find the total
surface area of the box.

    a. area of top ___________            b.  area of bottom ____________
    c. area of one end _________         c.  area of the other end _________
    d. area of the front ________          e.  area of the back ___________
        Total surface area of the box = ______________ sq. in.

10.  Write a formula for finding total surface area of a rectangular prism. __________________

11.  If the box had a triangular base, how would you find the total surface area?

12.  What changes would you make if the base was a circle?

13.  How would you explain calculating total surface area to someone?
 
 
 
 

Name __________________                  Date  _______________________
 
 

Find the surface area of each prism; show your work and formulas used.  You may wish to draw a sketch or net of the prism.

1. A box with each side measuring 8 feet.

2. A rectangular prism 7" wide, 9" long and 14" tall.

3. A triangular prism with base 6 meters long and 2.5 meters high with an overall height of 4 meters.

4. A rectangular prism 9cm by 12cm by 9cm.

5. A triangular prism with base sides 20m and 13m, triangular height 12m and overall height 15 m.

6. A triangular prism with base sides 10 yd by 10yd, triangular height 18 yds and overall height 15
yds.

7. A triangular prism with right triangle 24" by 25" whose height is 7".

8. A rectangular prism 2mm wide by 15mm long and 1mm high.

9. A rectangular prism 12' by 8' by 16 1/4 '.

10. A compact disc case with length 14.3cm, width 12.5cm and height 1cm.

from Mathematical Connections by Houghton Mifflin.
 
 

       Name __________                                                Date _________
 
 

1. Can you find a relationship between the number of faces, vertices, and edges of a polyhedra?
2. Write a formula using F for faces, V for vertices, and E for Edges.  __________________.
3. In the last column of the above table, substitute the values for the different polyhedra and see if your formula always works.
4. Access Euler and Polyhedra   to learn more about Euler and the five Platonic Solids.
5. Write a short paragraph about Euler and what you learned about the Platonic Solids.

Name __________________                  Date  _______________________

     Companies must pack and transport their products in shipping containers.  Your teacher has collected
a variety of balls.  You will work with your group to design, draw, and build a shipping box for the ball
you are given.  Make sure that the ball fits snugly into the box and that no stuffing is needed.  Record
measurements, create a net (drawing of your box showing all sides), and work with your group to build
the box.  Complete this worksheet about your work, show formulas, and write a summary paragraph as
detailed below.

1.  Measure and record the dimensions of the ball.____________________________________

2.  Write the formula and calculate the circumference of the ball. ________________________
      _________________________________________________________________________
     __________________________________________________________________________

3.  How long, tall, and wide must your box be to fit the ball?  __________________________

4.  What is the surface area of each face of the box?  _________________________________
     _________________________________________________________________________
     _________________________________________________________________________

4.  What is the volume of the box?  _______________________________________________
     _________________________________________________________________________

5.  What part of the volume of the box is not occupied by the ball?  ___________________________
     _________________________________________________________________________

6.  What tools did you use to measure dimensions of the ball?  Why did you use these?  _____
     _________________________________________________________________________

7.  How would you modify this box to contain three balls the same size as the one ball you have?
________________________________________________________________

8.  If you wanted to ship a ball shaped like the one you have but twice as tall, how would you modify
your shipping box? _______________________________________________________

9.  Would you want more or less volume in a shipping box to ship a fragile odd-shaped item?  Explain the reasons for your decision.  __________________________________________________________

10. Write a paragraph explaining how your group determined the size and shape of the shipping box and what you learned about surface area in this lesson.
 
 
 

Name  _______________                                                Date _______________

    Write the formulas your class has developed.  If there are any that you don't understand, ask members of your group or the teacher for help.  Then locate at least 3 cylinders in your classroom and complete the following worksheet.  Remember to check your calculations carefully.
 
 

1. With your group, measure the diameters and height of the two listed cylinders.
2. Write formulas and calculate the circumference and area of the bases.
3. Write formulas and calculate the surface area of the cylinders.
4. Write formulas and calculate the volume of the cylinders.
5. Find at least 3 other cylinders in the classroom (or brought from home).  Complete the chart for each of these cylinders.
6. Remember to check your work with a teammate or trade your three cylinders with those of another group and check each other's measurements and calculations.
7. Write a paragraph explaining what you learned about finding surface area and volume of prisms and cylinders.  How are the procedures alike;  how are they different?

 

Quiz - Surface Area and Volume of Cylinders

Name __________________    Date  __________________
 
 

Write the appropriate formula and find the surface area and volume of each cylinder.  You may want to draw the net for the cylinder to help you find the surface area.  Show your work and label your answers appropriately.

1. A cylinder whose diameter is 12 yd and whose height is 16yd.

2. A tank 70 mm long that has a 28mm diameter.

3. A pipe 9m in diameter that is 22.5m long.

4. A water tower 15 m in diameter that is 20 m tall. .

5. A semicirclular tunnel that has a radius of 30 ft and a length of 800 feet.
 
 
 

 
    Name ___________________    Date  __________________

    For these problems:

1. Write the appropriate formula
2. Show steps in calculations
3. Write answer with appropriate label, i.e. ft.sq, ft. cubed, etc.
1.  How much contac paper would be needed (without overlap) to cover a rectangular box 30 inches long, 14 inches tall, and 23 inches wide.

2.  Find the volume of the rectangular box in problem

3.  Calculate the surface area of a silo whose radius is 12 feet and whose height is 42 feet.

4.  How much grain would the silo in problem #3 hold?

5.  If a gallon of paint covers 180 square feet, how many gallons would be needed to paint a water19 foot tall reservoir whose diameter is 72 feet?

6.  Tobbler candy boxes are triangular prisms.  The Tobbler Company has a candy box outside itshome office that is 38 feet tall.  If the base of the container is 75sq ft., what is the volume of thebox?

7.  If the radius of a can of coffee is 5 inches and the can is 11 inches tall and contains 39 oz of coffee, how much coffee is contained per cubic inch?

8.  What is the volume of a 12 foot tall triangular prism whose base is 7 meters and height is 9 meters?

9.  Find the surface area of a rectangular prism 15 cm long, 17 cm high, and 7 cm wide.

10. What is the volume of the rectangular prism in #9?
 
 
 
 
 
 
 
 

Answers:
    1.  Total surface area = 2 top + 2 end + 2 side
                                       2(30 x 14) + 2(14 x 23) + 2(23 x 30)
                                        840          +      644      +    1380    =    2864 square inches
    2.  Volume = l w h
                        30 x 23 x 14 = 9660 cubic inches
    3.  Total surface area = 2 circles = (2pi r²) + rectangular sides if cylinder is split from top to bottom
         = (2 pi r h)    2(3.14)(12)(12)     +  2(3.14)(12)(42)
                             904.32                +   3165.12             =      4069.44 sq. ft.
       **note:  interior of bottom is accessible, exterior would not be
        accessible since the silo is sitting on the ground.
    4.  Volume = area of base x height  =  pi x r² x h
                                                        (22/7)(12)(12)(42) = 19,008
    5.  Surface area = pi x r² x h
                               2(3.14)(36)(36)(19)  = 154638.72      = 859.1 gal
                                        180                        180

    6.  Volume = base (1/2 bh) x height (h)
                                    75    x   38 = 2850 cu. ft.

    7.  Volume = pi x r² x h
                   3.14 x 5 x 5 x 11      =  863.5   =  22.1 oz/cu in
                                39                        39

    8.  Volume = 1/2 bh x height
                        1/2(7)(9)(12)  = 378 cu. meters

    9.  Surface area = 2 top + 2 side + 2ends
                               2(15)(17) + 2(17)(7) + 2(7)(15)
                                 510       +      238   +      210   =   958 sq. cm.

    10.  Volume = l w h
                         (15)(7)(17) = 1785 cu. cm.
 
 



Rubric for Surface Area - Shipping Box Activity



 
 
 
 
 
 
 
 
 
 
 

	Formula	Calculations	Written Summary
4	Correct formulas are applied.	Correct formulas are applied.	Written summary fully explains procedures.
3	Formulas are correct and most have been applied correctly.	Most calculations are complete and correct.	Written summary provides accurate explanation of most procedures.
2	Formulas are     correct; application  lacks completion or contains some error.	Calculations are complete, but may contain some errors.	Written explanation is provided but does  not fully explain some of the procedures.
1	Formulas are either lacking or are inaccurate.	Calculations are inaccurate or absent.	Either no explanation is provided or explanation lacks essential information.

    ***A score of 1 or 2 in any area indicates the need for reteaching content associated with the related material.