Pin it!
Google Plus

Are They Possible?

  • Lesson
6-8
1
Geometry
Unknown
Location: Unknown

Students examine some isometric drawings that seem to be impossible and investigate one way Escher used to create these "impossible" figures.

In previous lessons, students saw that isometric drawings were not always what they appeared to be. A Dutch artist, M.C. Escher (1898-1972), is famous for his use of unusual perspectives to trick the viewer into seeing "Impossible Figures." In this lesson, students will examine some isometric drawings that seem to be impossible, and they will investigate one way Escher used to create these "impossible figures."

Project the following images for the students to see:

A.2037 escher1 
B. 2037 escher2 
C. 2037 escher3 

As in previous lessons, you may print out the following PDF to create an overhead transparency:

1110 overhead  Are They Possible? 

Students should attempt to mentally construct each figure before using the isometric drawing tool.

Now, using the Isometric Drawing Tool , students should build each figure. Next, they should use the View feature 1983 eye to look at the figure from different perspectives. Students can open several tools so they can have access to all three drawings.

Now students should try the following:

  • Choose one of the three figures.
  • Delete a single cube from the picture, and use the View feature to explore the results.
  • Recreate the image and delete a different cube. Try this several times with each drawing.

Students can repeat with the other figures and discuss observations with a partner. Alternatively, students can create "impossible figures" and ask a partner to repeat this activity.

 
none

Questions for Students 

  1. Why do you think some people call these figures impossible? 
  2. What about isometric drawings creates these false impressions? 
  3. Do you think it is ever possible to have an isometric drawing that does not represent any 3-dimensional object? If so, can you draw one either on paper or using the applet? If not, can you explain why any isometric drawing created by the drawing tool is some 3D shape? 
 
1983icon
Geometry

Exploring the Isometric Drawing Tool

6-8
Students explore using the isometric drawing tool and gain practice and experience in manipulating drawings.
2009icon
Geometry

Finding Surface Area and Volume

6-8
Using the isometric drawing tool, students build three-dimensional figures and find the surface area and volume of each figure.
2017icon
Geometry

Building Using the Front-Right-Top View

6-8
Students explore drawing the front-right-top view when given a three dimensional figure built from cubes. Students also explore building a three dimensional figure when given the front-right-top view.
4024icon
Geometry

Using Cubes and Isometric Drawings: Mat Plans

6-8
Students explore drawing a mat plan when given a three dimensional figure built from cubes. Students also explore building a three dimensional figure when given the mat plan.
2028icon
Geometry

Do They Match?

6-8
Using three dimensional figures they have constructed, students determine when two isometric drawings can represent the same shape and explain their reasoning. Students will also determine what possible shapes might have the same isometric drawing and explain their reasoning.

Learning Objectives

Students will:
  • Examine isometric drawings that seem to be impossible
  • Investigate one way Escher used to create these figures
 

NCTM Standards and Expectations

  • Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

Common Core State Standards – Mathematics

Grade 7, Geometry

  • CCSS.Math.Content.7.G.A.2
    Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.