Abstract Expressionism – Measurement Perimeter, Area and Volume

Grade Level: 7
Classroom Time: Two 60 minute class periods

Materials Needed:

Metric and Customary ruler
Protractor
Pencil
Handouts
Paper or card stock
Cube, cylinder, pyramid, cone
Blank Paper or cardstock
Ziploc bags of 24 unifix cubes; one bag per student (optional)
Markers and/or map pencils

Objectives:

TSW: Discuss units used based on what is being measured
TSW: Discuss the use of a protractor
TSW: Measure angles and lengths using metric and customary measurement and find the
circumference, perimeter and area
TSW: Examine and discuss Abstract Expressionism
TSW: Discuss finding the volume of a prism
TSW: Review volume formulas for cylinders, cones, and pyramids
TSW: Examine and review volume
TSW: Examine the difference of areas
TSW: Explore the difference of volumes
TSW: Create an Abstract Expressionism art piece

Session 1: Angles, Perimeter, Circumference and Area

Briefly review the two measurement systems: customary and metric, and the value of using the appropriate unit. Review area, perimeter, diameter and circumference
1. 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 the desk using appropriate units
  - The length and height of the wall – meters, decimeters, feet, yards
  - The estimated area and perimeter of the wall using appropriate units
  - The diameter of the clock face – inches, centimeters
  - The estimated circumference and area of the clock
  - The estimated angle measurement between the two hands of the clock
  - Continue as needed
Briefly review types of angles and finding their measure using a protractor through the use of an overhead projector and various angles including acute, obtuse and right angles or one of the following websites. (Image of a protractor and various angles are available on the optional handout, continue as needed.)
1. Website support
Students review measurement – Handout 1 - Abstract Blues and Windows Review, finding the lengths of sides, angle measurement, perimeter, circumference and area
Explore and define Abstract Expressionism –
1. Websites for information and examples of Abstract Expressionism:
  1. www.artlex.com/ArtLex/a/abstractexpr.html
  2. http://www.artcyclopedia.com/history/abstract-expressionism.html
2. Additional examples of Abstract Expressionism:
After viewing the above examples of Abstract Expressionism discuss the different types of pieces the students observed. “What types of emotions did they observe?”, “What types of geometrical forms did they observe?”, “Did some appear to be more consciously created vs unconsciously created?” etc.

Session 2: Volume and Expression

Briefly review volume. Using the same size pieces of cardstock or paper, create a pile. Discuss how the pile of cardstock or paper is getting taller. As the pile gets taller volume is being created. Finding the area of each sheet and multiplying it by the number of sheets provides the volume, ignoring the width of the cardstock/paper. Review formulas using visuals of each object. If actual objects are not available create using a piece of paper or cardstock.

Optional Activity:
Give each student 24 unifix cubes. The students work in groups of 3 to construct as many different rectangular prisms as they can that have different volumes, not all cubes have to be used. Make a table of the class’ findings. Connect the formula (l x w x h) for finding volume to the models.

Website Support:
1. Formulas - http://www.factmonster.com/ipka/A0876212.html, http://www.infoplease.com/ipa/A0876212.html
2. Sample problems - http://go.hrw.com/resources/go_sc/hst/HSTMW301.PDF, http://boston.k12.ma.us/teach/technology/select/grade7/units/
  worksheets/fw/3_1_FindingVolume.doc
Students review finding the volumes of cubes, distribute Handout 2
Display http://nmaa-ryder.si.edu/images/1971/1971.306_1b.jpg, “Night Glow” by Adolph Gottlieb. Point out the red circle that appears “inside” the white circle. Ask the students how they would find the area of the white “rim.” Objective: Subtract the area of the red circle from the area of the white circle, A(red) – A(white) = A(rim). (Handout with picture located with optional handouts.)

Display http://americanart.si.edu/images/1986/1986.6.37_1b.jpg, “Hibernation” by Morris Graves. Point out the two red areas on the circle along with the white areas. Have the students work in pairs to find the “total” red areas, two parts, and find the “total” white areas, the inner circle and the middle rim. Share observations with the class and their process in finding the areas. (Handout with picture located with optional handouts.)
Display http://www.noguchi.org/redcube.htm, “Red Cube” by Isamu Noguchi. Point out the “hole” in the sculpture. Discuss the shape of the sculpture and the shape of the “hole,” cube and cylinder. Discuss how to find the volume of the cube, the volume of the “invisible” cylinder, and the volume of the cube with the cylinder removed.

Give the students the following dimensions for the figure. (Have the students work in pairs to find the volumes discussed above.)

Cube: Length, width and height = 15.5 m
Cylinder: Height – 15.5 m
Diameter – 5 m

Volume of the Cube – Volume of the Cylinder = Volume of the Red cube
Optional: Using the information found above and with the students still working in pairs have the students decide how they would find the weight of the “Red Cube” if it was made of Aluminum, oxide which weights 1522 kg/cu m.
Students will create an abstract Expressionism art piece. It can either be created on paper or constructed with various supplies. The art pieces can be displayed as an art gallery in a corner of the room. Using the individual student art pieces students will discuss and find various angles, dimensions, perimeters, areas and volumes as applicable.

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:

Understand numbers, ways of representing numbers, relationships among numbers, and number systems

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

Understand meanings of operations and how they relate to one another

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

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:

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.

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.

Knowledge and skills.
1. Perception. The student develops and organizes ideas from the environment. The student is expected to:
  1. illustrate ideas from direct observation, imagination, personal experience, and school and community events,
1. Response/evaluation. The student makes informed judgments about personal artworks and the artworks of others. The student is expected to:
  1. analyze and compare relationships; such as function and meaning, in personal artworks;

Abstract Blues/Windows Basic Review
Hand out 1 Continued
Red and Blue Composition – Ad Reinhard

Abstract Blues

Find the diameter of the circle in both customary and metric measurements.

d = ___________ (label) d = ___________ (label)

Find the circumference of the circle using both diameters.

c = ___________ (label) c = ___________ (label)

Find the measures of the following angles:

m<1 = _______ m<2 = _______ m<3 = _______ m<4 = _______

m<5 = _______ m<6 = _______ m<7 = _______ m<8 = _______

List the acute angles. _____________________________________________

List the obtuse angles ____________________________________________

List the right angles. _____________________________________________

“The Windows” – John Ferren

The Windows

Find the lengths of the sides of figure one using both customary and metric units. Label

Customary: l = _____, w = ______ Metric: l = ______, w = ______

Do the same for figure 2.
Label

Customary: l = _____, w = ______ Metric: l = ______, w = ______

Find the perimeter of figure one and figure 2 using both units. Label

Customary: P = ______ Metric: P = ______

Find the area of figure one and figure 2 using both units. Label

Customary: A = ______ Metric: A = ______

Structured or Empty Expressions – Volume Review
Handout 2
(Small Portion of Distorted Cubes (A-E))

Sol Lewitt

SolLeWitt
Distorted Cubes

Estimate the perimeter and area of each visible side (3 each) of the three figures. Label

Rectangular solid 1 – Perimeter _____________        Area ______________________

                          Perimeter _____________        Area ______________________

                          Perimeter _____________         Area ______________________

     Rectangular solid 2 - Perimeter _____________          Area ______________________

                        Perimeter _____________        Area ______________________

                        Perimeter _____________        Area ______________________

     Rectangular solid 3 - Perimeter _____________          Area ______________________

                        Perimeter _____________        Area ______________________

                        Perimeter _____________         Area ______________________

Estimate the volume of each rectangular prism, label:

V1 = ___________ V2 = ______________ V3 = _______________

Structured or Empty Expressions – Volume Review
Handout 2 Continued
(Small Portion of Two Figures (Menhirs))

Abstract

Find the circumference of each circle. Label.

C1 = ________________ C2 = ________________

Why doesn’t it appear to be an actual cylinder?

Find the volume of the cylinder using the diameter of the largest circle, C2. Label

V = __________________

Presuming it is a cylinder with an actual diameter of 8.2 cm and an actual height of 6.3 cm, find the volume. Label.

V = __________________

The artists had 168.432 cubic meters of steel to create the entire art piece. Estimate the amount of steel he saved creating the cylinder described above. Label.

Protractor - Optional

Angle Measurement
Class Review
Abstract

Find the measure, in degrees of each angle.

Abstract

Optional Handout
“Night Glow”

Abstract

Optional Handout
“Hibernation”

Abstract

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.