Utilization
Strategies 
DIMENSIONAL
SPACE: VARIOUS PERSPECTIVES ITV SERIES LIFE BY NUMBERS: SEEING IS BELIEVING GRADES 7  10 OVERVIEW This unit is an exploration of dimensional concepts, such as methods for interpreting 3D forms on a 2D surface, through the mastery of Linear Perspective. Students will learn how to recognize and create 3D forms on paper, understanding the use of angles and points to create the illusion of space in projected geometry. This basic understanding of Perspective will be extended by challenging the students to use their understanding of 3D on 2D to make a representation of a 4D form on paper. Finally, the students will be encouraged to imagine future applications of their knowledge of perspective. 

LEARNING OBJECTIVES Students will be able to demonstrate their enhanced spatial sense and 2D/3D visualization skills by creating easily recognizable models of forms. They will be able to identify depth cues in a picture or other 3D representation of 3D space, and will apply these reasoning skills to solve the more theoretical problem of 4D space through simple drawings of the hypercube. (National Math Curriculum Standards 3, 9 and 10)

MATERIALS
LECTURE/DISCUSSION Linear Perspective 



PREVIEWING ACTIVITIES Day 1 Briefly introduce Linear Perspective as a scientific method developed in the Renaissance (1500's CE) by important artists (Artist and Architect Brunelleschi, Leonardo Da Vinci, and Michelangelo). DAY 2 Review the Vanishing Point, and how it helped us to draw boxes. Ask the students to imagine how we could use the same technique of using verticals and diagonals to represent a 4D form on a 2D surface. Day 3 
FOCUS FOR VIEWING FOCUS FOR VIEWING Day 1 Day 2 Day 3 ASSESSMENT Student work should be evaluated as a small portfolio entitled Dimensional Space: Various Perspectives. Participation, effort, and accuracy should all be considered. ACTION PLAN Arrange a field trip to a prominent art gallery displaying Renaissance Artwork, and/or a scientific discovery museum in the area that explores the realm of Virtual Reality. Invite an Art Historian or Virtual Reality expert to speak to the class. 



VIEWING ACTIVITIES Day 1 Day 2 Questioning: How does an Xray present an unconventional perspective? How did Picasso's painting represent an unconventional perspective? · START viewing (after fastforwarding ~ 2 minutes) just before professor Tom Bancoff says, "When we saw the Hypercube for the first time…" STOP viewing after we see the projected shadow and Danny Glover says, "…In the same way, professor Bancoff can study the behavior of a Hypercube by watching it's 3D shadow. Day 3 EXTENSIONS Students can be encouraged to research and write up a brief summary of the development of Linear Perspective, as a language arts interdisciplinary activity. http://www.mcm.edu/academic Students could work cooperatively with an art class to create environments using 1 and 2 Point Perspective. 
POST VIEWING ACTIVITIES Day One: Day Two: Create a 3D Box using 2 Pt Linear Perspective on notebook paper. (Note: When drawing in 2 Pt Perspective, start with a line the corner edge of the cube and not a rectangle.) Day Three:
By Placing the Overhead Transparency (given to each student) on a viewing plane of a picture, (window, screen picture surface) and tracing the vertical lines and the "horizontal" lines, the students will isolate the Linear Perspective elements in the view. Note: If using a window as your plane, it is very important to stand in the same spot (without moving your head) as you draw the entire picture. Algebra Extension: Place the transparency (complete with important lines) on a sheet of graphing paper. Determine the Pointslope equations for each line by plotting points, and predict the Vanishing Point (by setting two equations equal to each other).Or: Using a large piece of paper, extend all lines to their vanishing point(s) with dotted lines to distinguish them from the structural elements of buildings. It works if you've traced carefully! Discuss student experiences involving Virtual Reality and its dimensions.


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