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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

When Negative is Positive – Perimeter, Area and Volume

Grade Level: 7
Classroom Time: Two 60-minute sessions

Materials Needed:

• Metric and Customary ruler
• Pencil
• Centimeter grid paper
• Handouts
• Blank Paper or cardstock
• Markers and/or map pencils
• Ziploc bags of 24 unifix cubes; one bag per student
• 6 – 10 cylindrical containers with lids, e.g. Pringle’s cans, lemonade powder container, etc.

Note: cover any volume or weight measurements on the cans

Objectives:
                                                             
TSW: Discuss units used based on what is being measured
TSW: Measure length, perimeter, and area in the customary and metric systems
TSW: Examine and define negative space
TSW: Examine and define Minimalism
TSW: Examine rectangular prisms and find their surface area and volume.
TSW: create a piece of art incorporating minimalism

Session 1: Area & Perimeter

1. Display the triangle here: http://www.artlex.com/ArtLex/c/Images/clozyr.3.gif on the overhead. Ask: What shape do you see? Students should see an implied triangle emerge. The triangle is in the negative space of the picture. The orange lines are in the positive space. What terms do we use to measure a triangle? (area, perimeter, inches, cm., length, width, height, etc.)

Briefly review the two measurement systems: customary and metric, and the value of using the appropriate unit. Review area, perimeter, diameter and circumference

Examples:

• The length of a pencil – cm or mm and inches
• The length of the side of a desk – decimeter, centimeters, inches, feet

The estimated area and perimeter of that same desk in appropriate units

• The length of height of the wall – meters, decimeters, feet, yards

The estimated area and perimeter of that same wall in appropriate units

• The diameter of the clock face – inches, centimeters

The estimated circumference and area of that same clock face

2. Examine two pieces of art with an emphasis on negative space. Identify the positive and negative space in each.

•  “Two Figures (Menhirs)” - sculpture
• “Butterfly” - serigraph

3. Distribute Handout 1 – Hidden Areas – Negative Space

Students work in pairs to find the sides, height, perimeter and area of the implied (negative space) triangle displayed on the overhead (number 1). Check for understanding. Students do the remainder independently.

Optional Follow-up: Write three problems for area and/or perimeter using this piece of art by Johannes Itten – display on overhead, handout for students.  http://www.moma.org/collection/provenance/items/images/1028.83.jpg
Switch problems with a friend and solve.

Session 2: Volume

1. Give each pair of students’ one rectangular prism (ex. Kleenex© box, Pop-Tarts© box, etc.), each a different size. Take the box apart and estimate the surface area in square centimeters. Give each pair a piece of centimeter grid paper to verify.
2. Give each student 24 unifix cubes. The students work in groups of 4 – 5 to construct as many different rectangular prisms as they can that each have a volume of 24 cubes: 6x2x2, 8x3x1, 24x1x1, 6x4x1, 12x2x1, 24x1x1. What is the surface area of each one? The volume? Make a table of the class’ findings. Connect the formula (LxWxH) for finding volume to the models. How are the surface area and the volume related?

Rectangular Prism #	Dimensions	Surface Area	Volume

3. Rectangular prisms as well as other geometric forms are a frequent subject in an art style known as “Minimalism.” In minimalism, the subject is reduced to only the most essential elements. Simple lines, shapes, forms and colors are important in minimalism. For more information see http://www.artlex.com/
4. Examine: “Equivalent VIII” by Carl Andre: http://www.tate.org.uk/servlet/ViewWork?workid=508&roomid=3669
   Discuss the art elements and principles important in this artwork: form, line, balance, unity. Is this art? Why or why not? Handout 2 – Area and volume
   
   Examine “Untitled” by Donald Judd
   http://www.tate.org.uk/servlet/ViewWork?workid=7770&searchid=10078&roomid=3669&currow=6&maxrows=11 – Handout 3 – Area and volume
5. Create small group minimalistic artworks. Give some parameters to make the activity more challenging; i.e. the art must fit in a five foot square space, and contain between 30 and 40 right angles. It must be comprised only of rectangular prisms.

Collect different sized boxes (cereal, tissue, etc.) and allow small groups to construct a group piece of minimalism.  Spray painting the boxes adds a nice finish to the pieces. Find the surface area and volumes of different boxes within the artwork.

Website to assist with measurement unit: http://www.learner.org/resources/series184.html?pop=yes&vodid=836316&pid=1992#

National Math Standards:

Geometry Standard:
A.) Analyze characteristics and properties of two- and three-dimensional geometric shapes
     and develop mathematical arguments about geometric relationships

•	precisely describe, classify, and understand relationships among types of two- and three-  dimensional objects using their defining properties;
•	understand relationships among the angles, side lengths, perimeters, areas, and volumes of    similar objects;

B.) Use visualization, spatial reasoning, and geometric modeling to solve problems use
     two-dimensional representations of three-dimensional objects to visualize and solve
     problems such as those involving surface area and volume;

•	use two-dimensional representations of three-dimensional objects to visualize and solve  problems such as those involving surface area and volume;
•	recognize and apply geometric ideas and relationships in areas outside the mathematics  classroom, such as art, science, and everyday life.

Numbers and Operations Standard:

A.) Understand numbers, ways of representing numbers, relationships among numbers, and number systems

•	work flexibly with fractions, decimals, and percents to solve problems;

B.) Understand meanings of operations and how they relate to one another

•	understand the meaning and effects of arithmetic operations with fractions, decimals,  and integers;

C.) Compute fluently and make reasonable estimates

•	select appropriate methods and tools for computing with fractions and decimals from  among mental computation, estimation, calculators or computers, and paper and  pencil, depending on the situation, and apply the selected methods;

Measurement Standard:

A.) Understand measurable attributes of objects and the units, systems, and processes of measurement

•	understand both metric and customary systems of measurement;
•	understand relationships among units and convert from one unit to another within the     same system;
•	understand, select, and use units of appropriate size and type to measure angles,  perimeter, area, surface area, and volume.

B.) Apply appropriate techniques, tools, and formulas to determine measurements

•	select and apply techniques and tools to accurately find length, area, volume, and  angle measures to appropriate levels of precision;
•	develop and use formulas to determine the circumference of circles and the area of    triangles, parallelograms, trapezoids, and circles and develop strategies to find the  area of more-complex shapes;
•	develop strategies to determine the surface area and volume of selected prisms,  pyramids, and cylinders;

 

TEKS – Grade 7 Math

7.2) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, or divides to solve problems and justify solutions.
     (B) use addition, subtraction, multiplication or division to solve problems involving fractions and decimals
     (F) select and use appropriate operations to solve problems and justify the selections
     (G) determine the reasonableness of the problem

7.4) Patterns, relationships and algebraic thinking. The student represents a relationship in numerical, geometric, verbal, and symbolic form.
      (A) generate formulas involving unit conversions, perimeter, area, circumference and symbolic form

7.8) Geometry and spatial reasoning. The student uses geometry to model and describe the physical world
      (C) use geometric concepts and properties to solve problems in fields such as art and architecture

7.9) Measurement. The student solves application problems involving estimation and       measurement.

      (A) estimate measurements and solve application problems involving length (including perimeter and circumference) and area of polygons and other shapes;
      (B) connect models for volume of prisms (triangular and rectangular) and cylinders to formulas of prisms (triangular and rectangular) and cylinders; and
      (C) estimate measurements and solve application problems involving volume of prisms (rectangular and triangular) and cylinders

7.13) Underlying processes and mathematical tools. The student applies Grade 7 mathematics to solve problems related to everyday experiences, investigations in other disciplines and other activities in and outside of school.
      (D) select tools such as real objects, manipulative, paper/pencil, and technology or techniques such as mental math, estimation, and number sense to solve problems

National Visual Art Standards

2. Using knowledge of structures and functions

• Students generalize about the effects of visual structures and functions and reflect upon these effects in their own work
• Students employ organizational structures and analyze what makes them effective or not effective in the communication of ideas

3. Choosing and evaluating a range of subject matter, symbols, and ideas

• Students integrate visual, spatial, and temporal concepts with content to communicate intended meaning in their artworks
• Students use subjects, themes, and symbols that demonstrate knowledge of contexts, values, and aesthetics that communicate intended meaning in artworks

6. Making connections between visual arts and other discipline

• Students compare the characteristics of works in two or more art forms that share similar subject matter, historical periods, or cultural context
• Students describe ways in which the principles and subject matter of other disciplines taught in the school are interrelated with the visual arts

TEKS - §117.35. Art, Grade 7.

C.  Knowledge and skills. 

(1) Perception. The student develops and organizes ideas from the environment. The student is expected to:  
(A) illustrate ideas from direct observation, imagination, personal experience, and school and community events,

(4) Response/evaluation. The student makes informed judgments about personal artworks and the artworks of others. The student is expected to:                                                              
(A) analyze and compare relationships; such as function and meaning, in personal artworks;

HISD CLEAR Objectives

ART.7.1.01: Develop perception skills necessary for generating and organizing four original art creations.
ART 7.2.10: Select significant artworks and styles, movements, and themes through four primary sources, visuals, museum study trips, and community resources.
ART 7.3.01: Develop an understanding of each of the types of visual expression in four, two- and three-dimensional visual artworks.
ART 7.3.02: Create both formal (symmetrical) and informal (asymmetrical) designs and compositions in four artworks.
ART 7.3.15: Analyze four unfamiliar artworks.

Hidden Areas – Negative Space
Handout 1

1. State the metric unit used to accurately measure the sides of the hidden implied triangle. ____________
   
   
   
2. State the customary unit used to accurately measure the sides of the hidden implied triangle. ____________
   
   
   
3. Measure the sides of the hidden implied triangle using both systems. Label.
   
   Metric _____________________                 Customary ________________
   
   
   
4. Find the height of the hidden implied triangle using both measurement units.
   
   Metric _____________________                  Customary ________________
   
   
5. Find the perimeter and area of the red and hidden implied triangles.
   
   Perimeter: Metric _________________         Area:  Metric ______________
   
   Customary _________________             Customary ______________

Hidden Areas – Negative Space - Continued

 

6.  Find the length of the side of the implied square using both measuring systems.
   
   Metric ____________________         Customary ___________________
   
   
7. Find the perimeter and area of the implied square.
   
   Perimeter:         Metric _________________         Area:  Metric __________________
   
   
           Customary _________________             Customary ______________
   
   
8. Find the diameter one of the implied circles.
   
   Metric _____________    Customary _____________
   
   
9. Find the circumference of one of the implied circles.  
   
   Metric ___________    Customary ____________
   
   
10. Find the area of one of the implied circles.    

 

Metric ____________        Customary _____________

Hidden Areas – Negative Space - Continued

 

 

11. Find the length of one side of the hidden cube using both systems
    
    Metric ____________                 Customary ___________
    
    
12. Find the perimeter of one side.
    
    Metric ____________                 Customary ___________
    
    
13. Find the area of one side of the cube. (Find the surface area of the cube.)
    
    Area/Surface Area Metric ______/______   Area/Surface Area Customary _____/_____
    
    
14. Find the volume of the cube.
    
    Metric ____________                 Customary ___________
    
    
15. Estimate the diameter of one of the circles.
    
    Metric ____________                 Customary ___________
    
    
16. Find the circumference of the circle
    
    Metric ____________                 Customary ___________
    
    
17. Find the area of the circle.
    
    Metric ____________                 Customary ___________
     

 

Johannes Itten
Optional

 

 

1.

 

 

2. 

 

 

3.

 

 

 

 

“Equivalent VIII” by Carl Andre
 Handout 2

 

The actual dimensions of the “Equivalent VIII” piece are 127 mm x 686 mm x 2292 mm. All the bricks have an equivalent measure. The art piece is one brick tall, the width consists of 6 bricks and the length consists of 10 bricks.

1. Find the dimensions of one brick. _____________________________________________

 

2. Find the surface area of one brick. ____________________________________________
   
   

3. Find the volume of one brick. ________________________________________________

 

4. Find the surface area of “Equivalent VIII.” ______________________________________
   
   

5. Find the volume of “Equivalent VIII.”  __________________________________________

 

 

“Untitled” – Donald Judd
Handout 3

 

The actual dimensions of “Untitled” are 229 mm x 1016 mm x 787 mm, from the bottom of the first rectangular prism to the top of the tenth rectangular prism.

1. How many “negative space” rectangular prisms would there be to fill in the space between the prisms? __________

 

2. Find the dimensions of one rectangular prism. ___________________________________
   
   

3. Find the surface area of one rectangular prism. __________________________________

 

4. Find the volume of one rectangular prism. _______________________________________
   
   
   

5. Find the total volume of all the actual rectangular prisms. __________________________

 

6. Find the total volume of the “negative space” rectangular prisms. ____________________
   
   
   

7. When you add the two volumes together, does it equal the total volume of ‘Untitled?” _______

 

8. If Mr. Judd asked you to give “Untitled” a name to describe it, what name would you give it?
   
   Why? __________________________________________________________________

“Shading Cubes and Cylinders”
An Enlightening Exercise
Handout 3

Shade each solid according to the direction of light indicated by the arrow.

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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.