Utilization Strategies

Lesson Plans

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by Celia Doerann-George
University of Idaho


GRADES 7 - 10


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.


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)



  • Notebook paper
  • pencil
  • Ruler, protractor
  • Large white paper
  • Worksheet #1
  • Overhead and markers
  • Image(s) with buildings, (window, picture, image on a TV screen, or web image on computer screen)
  • Graph paper (Algebra Extension)


Linear Perspective
Vanishing Point
Horizon, and Orthogonal Lines
The 3D cube represented on a 2D surface


Day 1
Introduce basic concepts of the unit such as space, dimensions, 2D representations of 3D objects, and perspective. This should be discussion-based, and students should be encouraged to offer definitions for these concepts. Question students about any familiar depth-cues in art or pictures they have noticed. Draw a picture on the board demonstrating relative position, size and overlap, and ask students which objects they think are in front of the others, and why.

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).

Review important spatial concepts to consider when one tries to translate a 3D object into a 2D representation of that object. Depth cues, such as objects or parts of objects getting smaller and closer to the horizon line as they recede from the viewer are important to review.

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
Review concepts of dimensional space and projected geomety. Review and display student activities, discussing the illusion of depth achieved.




Day 1
To give students a specific responsibility while viewing the segment, tell them that the information in the segment will instruct us on how to use Linear Perspective to represent 3D Space on a 2D Picture Plane.

Day 2
To give students a specific responsibility while viewing, tell them the video will help them to learn how to draw a 4D form called a "Hyperbox."

Day 3
To give students a specific responsibility while viewing, ask the following questions, and encourage them to make a list in their notebook. What are some things you might apply this knowledge to in your future? Where/when do people use their knowledge of Perspective on a daily basis? Can you think of any career fields in which Dimensions and Perspective would play an important role?


Student work should be evaluated as a small portfolio entitled Dimensional Space: Various Perspectives. Participation, effort, and accuracy should all be considered.


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.


Day 1
Life By the Numbers Seeing is Believing Program, Segment 2, "Linear Perspective." START viewing at beginning of Segment 2. STOP viewing after professor Sam Edgerton says, "...the west got a jump on the rest of the world in terms of an Industrial Revolution."

Day 2
Life By the Numbers Seeing is Believing Program, Segments 4 &5, "Invisible Realities." START viewing at beginning of Segment 4. (X-ray, Picasso perspectives) STOP viewing at beginning of Segment 5. · Viewing time is approximately 3.5 minutes.

Questioning: How does an X-ray present an unconventional perspective? How did Picasso's painting represent an unconventional perspective? ·

START viewing (after fast-forwarding ~ 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
Life By the Numbers Seeing is Believing Program, Segment 7, "Worlds in Progress." START viewing at beginning of Segment 7 START viewing after Marcos Novak says, "…a vast unexplained territory that I think we can venture into and come back with incredible insight and beauty." (before he says, "One of the thrills…)


Students can be encouraged to research and write up a brief summary of the development of Linear Perspective, as a language arts interdisciplinary activity.


Students could work cooperatively with an art class to create environments using 1 and 2 Point Perspective.



Day One:
Create a 3D Box using 1 Pt Linear Perspective on ordinary notebook paper with pencil.
Geometry Extension: Measure and label all the angles in your drawing. What do you notice?

Day Two:
On large white paper, create a Room with floor boards, a window on one wall, a box on the floor against the other wall, and a box somewhere on the ceiling. (worksheet #1 Note: When drawing the grid for the floor boards, remember to use orthogonals. Do an example of a room with different boxes and floorboards.

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:
Window Drawing: Seeing the Linear Perspective.

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 Point-slope 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.


For additional lesson plans and ideas relating to this topic and many others try TeacherSource at PBS Online! You will find activities, lesson plans, teacher guides and links to other great educational web sites! Search the database by keyword, grade level or subject area! Mathline and Scienceline are also great resources for teachers seeking teaching tips, lesson plans, assessment methods, professional development, and much more!

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