EDITED Resources. The resources on this page have been aligned with the revised K-12 mathematics TEKS. Necessary updates to the resources are in progress and will be completed Fall 2006. These revised TEKS were adopted by the Texas State Board of Education in 2005–06, with full implementation scheduled for 2006–07.

Clarifying Lessons

Mathematical Models with Applications: Two-Point Perspective Drawing

activity under revision

Lesson Overview

Students use interactive geometry exploration software (GES) to create a two-point perspective drawing.

Mathematics Overview

Students explore basic geometry concepts in and geometric transformations of the model of a rectangular prism viewed from different perspectives.

Set-up (to set the stage and motivate the students to participate)

  1. Two-point perspective drawings are often used in an architect's illustration of a proposed building. The advantages are that two or three faces of the building are visible and the exaggerated perspective makes the drawing more realistic. Have students find illustrations that involve perspective. Save them so that later they can discuss if they think any of them are two-point perspective drawings.
  2. Two-point perspective drawings bring a host of geometrical considerations into a drawing. To begin such a drawing, have students choose a horizon line and two vanishing points on this horizon line (see Figures 1 and 2 on the Activity Sheet). The relationship of the horizon line to the object determines the point of view (see Question 2 on the Activity Sheet).
  3. All the "vertical" lines in the drawing must be perpendicular to the horizon line, so they all appear parallel to each other (see Figure 3 on the Activity Sheet). But the lines in the drawing representing the "horizontal" edges of the object all intersect at one or the other of the vanishing points, creating the illusion of depth (see Figures 4 and 5 on the Activity Sheet). All those intersecting non-vertical lines represent lines that are supposed to be parallel. Thus, these drawings are an excellent way to study parallel lines and corresponding angles.
  4. Instead of having each student draw more than one two-point perspective drawing in order to see how the placement of the horizon line and the position of the vanishing points affect the perspective, have students use an interactive GES package (such as The Geometer's SketchpadÆ) to create a two-point perspective drawing of a common rectangular box and then use the characteristics of the software package to manipulate the drawing in various ways (see the Activity Sheet).
  5. Monitor the students carefully to insure that the instructions are actually being followed so that the drawing behaves appropriately when different parts of it are manipulated.
  6. After the sketch is finished, have students:
    • change the relationship of the horizon line to the object by moving the vertical segments,
    • move the vanishing points closer together and farther apart and
    • record their observations as to how the rest of the drawing follows so that all of the rules used in the original construction remain valid.

Teacher Notes (to personalize the lesson for your classroom)

Guiding Questions (to engage students in mathematical thinking during the lesson)

  • What rules about the drawing remain the same when you change the relationship of the horizon line to the object or change the distance between the vanishing points? (All the vertical lines remain parallel to each other and all the non-vertical lines continue to intersect at one of the vanishing points.)
  • How can you make the horizon line appear high in relation to the object? (By lowering the object by translating one of its vertical segments down the page.) (M.1.A, B; M.9.A)
  • How can you make the horizon line appear low in relation to the object? (By raising the object by translating one of its vertical segments up the page.) (M.1.A, B, C; M.9.A)
  • What happens when the horizon line moves higher in relation to the object? (If the horizon line is high in relation to the object, the viewer sees the top of the object.) (M.9.A)
  • What happens when the horizon line is low in relation to the object? (Lowering the horizon line (by raising the object) brings the bottom of the object into view.) (M.9.A)
  • What happens when you change the distance between the two vanishing points? (The relative distance between the vanishing points determines the depth of the perspective; decreasing the distance between the vanishing points heightens the perspective effect.) (M.9.A)
  • What happens if you move one of the vertices of the rectangular prism? (M.9.A)
  • What happens if you move one of the non-vertical segments? (M.9.A)

Teacher Notes (to personalize the lesson for your classroom)

Summary Questions (to direct students' attention to the key mathematics in the lesson)

To encourage students to connect the directions for making the two-point perspective drawing to the geometric properties of the model, ask questions such as:

  • Why is it important to make the vertical lines in the object all perpendicular to the horizon line? (So they all appear parallel to each other in the object.) (M.9.A)
  • Why did we only construct three vertical lines? (Because that is the most a viewer can see of the object at any one time.) (M.9.A)
  • Why was it necessary to select both a point on the horizon line and the horizon line to construct a vertical line? (Because that is a minimum amount of information that needs to be provided to the computer in order to construct a line perpendicular to the horizon line). (M.9.A)
  • What other ways might you be able to construct a line perpendicular to the horizon line? (M.1.A, B, C; M.9.A)
  • How is constructing two perpendicular lines with the software different from just drawing two lines that appear to be perpendicular? Why are the construction steps important? (The properties of perpendicularity have to be satisfied and remain constant when other parts of the drawing are moved.) (M.1.A, B, C; M.9.A)
  • What other observations did you make in exploring with two-point perspective? (M.9.A)

Teacher Notes (to personalize the lesson for your classroom)

Assessment Task(s) (to identify the mathematics students have learned in the lesson)

Student observations about the transformations at the end of the Activity Sheet can be used for assessment.

Teacher Notes (to personalize the lesson for your classroom)

Extension(s) (to lead students to connect the mathematics learned to other situations, both within and outside the classroom)

  • Have students find illustrations that use two-point perspective and post them in the classroom for discussion.
  • Have students try to draw a two-point perspective of a triangular prism or rectangular pyramid.

Teacher Notes (to personalize the lesson for your classroom)