---

Project GRAD Houston
1510 Jensen Dr.
Houston, TX 77020
832-325-0325 (main)

Mailing address:
Project GRAD Houston
PO Box 15568
Houston TX 77220-5568

Space Abounds

Grade Level: 8
Classroom Time: Two 60-minute sessions

Materials Needed:

• Protractor
• Pencil
• Handout
• Blank Paper or cardstock
• Markers and/or map pencils
• Cardboard, glue, containers
• Different containers of various shapes
• Connecting cubes
• Graph paper/large graph paper
• 1” graph paper copied onto cardstock (attached or http://www.math.kent.edu/~white/graphpaper/one.pdf)
• Metric and customary measuring tool
• Scissors

Objectives:
TSW: Identify different types of solids
TSW: Identify lateral area, surface area and volume formulas for each solid
TSW: Examine and respond to Sculpture Architecture
TSW: Identify and find the lateral area, total surface area and volume of various solids
TSW: Review the Pythagorean Theorem and use it
TSW: Identify slant height
TSW: Find the volume of pyramids
TSW: Create a piece of architectural sculpture using at least 5 different three-dimensional  
         forms

1. Quick review/focus: As a class, students will review different shapes, prism, pyramids, cylinders, spheres and cones discussing how to determine their lateral area, total surface area, and volume as applicable. Different containers will be presented to the class for them to review, ie cereal box or shoe box, ball, Pringles® Can, etc…

2. Quick review/focus: A prism constructed with 9 connecting cubes is held up. Together as a class review the formulas and find the actual calculations of the dimensions, lateral area, total surface area, and volume. Discuss the difference between the total surface area and the volume. Find their ratios.

Class activity: Divide the class into small groups. Give each group 18 cubes. Assign each group a given prism to construct, 1x2x9, 1x3x6, 1x1x18. Have each group find the lateral area, total surface area, volume and create a net on graph paper of their designated prism. Create a class chart listing the above. Compare the volumes, discuss. Discuss the volumes if the measurements were changed from inches to yards, etc..

Display a picture of the Freedom Tower - http://skyscraperpage.com/cities/?buildingID=7788. Using the actual measurements find the lateral area, total surface area and volume of the base.( Width – 61.0 m, Height – 56.9 m, Length – 61.0 m) Each group creates a real world problem relating to a prism, teacher may assign prisms or students may select their own. Create a handout with student’s problems for every one to solve.

3. View examples of Sculpture Landscapes:
   • http://www.noguchi.org/redcube.htm
   • http://garden.walkerart.org/artwork.wac?id=1255 – there is a virtual tour labeled at the top and by clicking the red links on the right side different sculptures will appear
   • http://www.metmuseum.org/special/Goldsworthy/roof2004_more.htm
   • http://www.tate.org.uk/collection/T/T07/T07144_9.jpg
4. Elicit responses from students detailing their like or dislike of the sculptures. Have them describe what they are viewing. Why do they think an artist would create the piece(s)? Have they observed sculpture around Houston? Where?

5. Display Sol LeWitt’s Five Open Geometric Structures.  Name each individual structure. Review formulas for finding the lateral area, surface area and volume for the cube, pyramid and prism as if they were solids. Create nets for each figure. Discuss what the students feel would be their actual height, width, etc.. What measures would be used to measure them? (in., ft, yd, m, cm, mm, etc…)

6. Handout 1 – Students review lateral area, total surface area and volume of prisms and spheres.

7. Review the Pythagorean Theorem – Activity http://regentsprep.org/regents/math/fpyth/TActive.htm and http://regentsprep.org/regents/math/math-topic.cfm?TopicCode=fpyth  OR Handout 2

8. Review pyramids – How to find the volume by finding the volume of the related rectangular prism - http://www.mathematische-basteleien.de/pyramid.htm - Good explanation and display and review of pyramids

     Handout 3 – Display the Louvre Pyramid on the overhead - 
     http://en.wikipedia.org/wiki/Image:HPIM1258.JPG – Answer questions involving  
     pyramids

9. Construct an architectural landscape using at least five different forms.
1. On 1” graph paper copied onto card stock draw nets for each form - http://www.math.kent.edu/~white/graphpaper/one.pdf (graph paper or attached)
2. Construct and join forms
3. Find the measurements of each form.
4. Find the volume of each form.

8th Grade TEKS

(8.8)  Measurement. The student uses procedures to determine measures of three-dimensional figures.

The student is expected to:
(A)  find lateral and total surface area of prisms, pyramids, and cylinders using concrete models and nets (two-dimensional models);
(B)  connect models of prisms, cylinders, pyramids, spheres, and cones to formulas for volume of these objects; and
(C)  estimate measurements and use formulas to solve application problems involving lateral and total surface area and volume.

(8.9)  Measurement. The student uses indirect measurement to solve problems.

The student is expected to:
(A)  use the Pythagorean Theorem to solve real-life problems; and
(B)  use proportional relationships in similar two-dimensional figures or similar three-dimensional figures to find missing measurements.

(8.10)  Measurement. The student describes how changes in dimensions affect linear, area, and volume measures.

The student is expected to:
(A)  describe the resulting effects on perimeter and area when dimensions of a shape are changed proportionally; and
(B)  describe the resulting effect on volume when dimensions of a solid are changed proportionally.

(8.14)  Underlying processes and mathematical tools. The student applies Grade 8 mathematics to solve problems connected to everyday experiences, investigations in other disciplines, and activities in and outside of school.

The student is expected to:
(A)  identify and apply mathematics to everyday experiences, to activities in and outside of school, with other disciplines, and with other mathematical topics;
(B)  use a problem-solving model that incorporates understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness;
(C)  select or develop an appropriate problem-solving strategy from a variety of different types, including drawing a picture, looking for a pattern, systematic guessing and checking, acting it out, making a table, working a simpler problem, or working backwards to solve a problem; and
(D)  select tools such as real objects, manipulatives, paper/pencil, and technology or techniques such as mental math, estimation, and number sense to solve problems.

(8.15)  Underlying processes and mathematical tools. The student communicates about Grade 8 mathematics through informal and mathematical language, representations, and models.

The student is expected to:
(A)  communicate mathematical ideas using language, efficient tools, appropriate units, and graphical, numerical, physical, or algebraic mathematical models; and
(B)  evaluate the effectiveness of different representations to communicate ideas.

(8.16)  Underlying processes and mathematical tools. The student uses logical reasoning to make conjectures and verify conclusions.

The student is expected to:
(A)  make conjectures from patterns or sets of examples and nonexamples; and
(B)  validate his/her conclusions using mathematical properties and relationships.

National Math Standards

Instructional programs from prekindergarten through grade 12 should enable all students to--

Geometry

• analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships;
• specify locations and describe spatial relationships using coordinate geometry and other representational systems;
• apply transformations and use symmetry to analyze mathematical situations;
• use visualization, spatial reasoning, and geometric modeling to solve problems.

Measurement

• understand measurable attributes of objects and the units, systems, and processes of measurement;
• apply appropriate techniques, tools, and formulas to determine measurements.

Visual Arts TEKS/National Standards

117.38 (1)  Perception. The student develops and organizes ideas from the environment. The student is expected to:

 (B)  define a variety of concepts directly related to the art elements and principles, using vocabulary accurately.

117.38.(2) Creative expression/performance. The student expresses ideas through original artworks, using a variety of media with appropriate skill. The student is expected to:

(A)  create artworks integrating themes found through direct observation, personal experiences, and imagination;
(B)  apply design skills to communicate effectively ideas and thoughts in everyday life; and
(C)  select appropriate art materials and tools to interpret subjects or themes when producing drawings, paintings, prints, sculptures, ceramics, fiberart, photography/film making, and electronic media-generated art, traditionally and experimentally.

117.38.(4) Response/evaluation. The student makes informed judgments about personal artworks and the artworks of others. The student is expected to:

(A)  analyze with the teacher or peers personal artworks in progress, using critical attributes, and participate in individual and group critiques; and
(B)  analyze original artworks, portfolios, and exhibitions by peers and others to form conclusions about formal properties, historical and cultural contexts, intents, and meanings.

National Visual Arts Standards

• Choosing and evaluating a range of subject matter, symbols, and ideas.
• Reflecting upon and assessing the characteristics and merits of their work and the work of others.
• Making connections between visual arts and other disciplines.

 

 

A Walk in the Garden – Minneapolis Sculpture Garden
Handout 1

 

1. What shapes are depicted above? ________________________________

2. What title would you give the sculpture? ________________________

3. What metric measure would you use to find the length of the side of the actual figure? ______________________, Customary ________________

4. Using a pencil, extend the two sides and bottom of the middle prism in order to create a visual of the entire prism.

5. Measure the sides of the middle prism. _____________________________

6. Find the lateral area ___________________________________________

7. Find the total surface area ______________________________________

8. Find the volume ______________________________________________

9. If the length of the bottom side is 38 in and each side measures 18 inches, find the area of the front side of the prism. __________________

10. Given the dimensions above and a width of 36 in, find the total surface area. _____________________________________

11. Find the volume. _____________________________________________

A Walk in the Garden – Minneapolis Sculpture Garden
Handout 1 Continued

 

12. Find the ratio of the measured bottom of the prism to the given measurement. _____________________

13. Find the ratio of the volume of the measured prism to the volume using the given measurement. ____________________

14. Are the two ratios equivalent? Why or why not.
    
    

---

 

 

 

15. What is the shape of the red object? ___________________________________

16. State the formula for finding the volume of the red object? _________________

17. If the radius of the cherry is 40 inches, find the circumference _______________

18. Find the volume of the cherry. ________________________________________

19. Measure the diameter of the cherry in the picture. ________________________

20. Find the circumference of the cherry using your measurement. ______________

21. Find the volume of the cherry using your measurement. ___________________

22. Find the ratio of the measured diameter and the actual diameter. ____________

23. Describe the differences in the volume.
    
    

---

 

24. Select one of the above sculptures; create three questions involving lateral area, total surface area and volume.
    A.
    
    

---




---


B.



---




---


C.



---




---

What was Pythagoras Thinking About?
Handout 2

1. Cut out the following squares, 3x3, 4x4, 5x5, 6x6, 8x8, 10, 10, 12x12, 13x13, 15x 15, and 17x17 using graph paper.

 

2. 
   
   On a second piece of graph paper arrange the squares to form right triangles without overlapping the squares. Trace.

3. Label each side of the triangle on the graph paper using the length of the side of the square. How many right triangles can be created? ____________________

4. List them _________________________________________________________

5. Do you see a pattern? If so what is it?
   
   

---

6. State the Pythagorean Theorem. Use the above information to show the Pythagorean Theorem.
   
   

---

7. Check to see if the following are sides and the hypotenuse of right triangles. Circle Y or N.

 

9, 12, 15              Y        N                            5, 10, 15              Y        N

10, 24, 26            Y        N                             3, 4, 5                 Y        N

4, 5, 6                  Y       N                             7, 9, 14                Y        N    

 

Are all Pyramids In Egypt? – A trip to Paris, France
Handout 3

I.M. (Ieoh Ming) Pei
Pyramid of the Louvre

The above pyramid is the entrance to the Musée du Louvre (or The Louvre Museum) in Paris France

1. The base of the pyramid is a square with sides of 35 meters. Find the area of the base. ____________________ (Label)

2. State the Pythagorean Theorem. _____________________________________

3. The height of the pyramid is 20.6 m, find the slant height, the height of the triangular side (Pythagorean Theorem). ___________________________ (Label)

4. Find the area of the triangle. ___________________________________ (Label)

5. Find the lateral area of the pyramid. _____________________________ (Label)

6. Find the total surface area of the pyramid. ________________________ (Label)

7. State the formula for the volume of a pyramid. __________________________

8. Find the volume. ____________________ (Label)

9. A small pyramid is constructed in the middle of the above pyramid. Its dimensions are 1/10th the above, find the lateral area, total surface area and volume of the smaller pyramid. ________________________________________

10. Compare the two lateral areas, total surface areas and volumes. Discuss your findings. __________________________________________________________
    
    

---

---

 

 

---

The mission of Project GRAD is to ensure a quality public education for all students in economically disadvantaged communities so that high school and college graduation rates increase.