## Are They Possible?

- Lesson

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:

| C. |

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

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 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.

- Isometric Drawing Tool
- Are They Possible? (Images to be made into an overhead transparency)

**Questions for Students**

- Why do you think some people call these figures impossible?
- What about isometric drawings creates these false impressions?
- 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?

### Exploring the Isometric Drawing Tool

### Finding Surface Area and Volume

### Building Using the Front-Right-Top View

### Using Cubes and Isometric Drawings: Mat Plans

### Do They Match?

### Learning Objectives

- 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.