## Are They Possible?

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:

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

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 **Inspect **mode
to look at the figure from different perspectives. Students can open
several windows so they can have access to all three drawings.

- Find the surface area and volume.
- Sketch the front, right, and top (FRT) view.
- Sketch the mat plan.
- Discover if it is possible to have another figure that has the same isometric drawing.

You can either have students share their results or collect this as a form of assessment.

- Computers or tablets with Internet access
- Are They Possible? Overhead

**Assessment Options**

- Collect students' work as a form of assessment.
- Have students pair up and check each other's answers (for the last exercise of this unit). Circulate the room and take note of students who are struggling to remember previous lessons.

**Extensions**

- Have students research M.C.Escher and his work. Have them write a paragraph on how isometric drawings are useful in art.
- A great lesson to follow up this unit is
*Hotel Snap*, where students will be required to build a high-profit yielding hotel using snap cubes. This lesson will take approximately three days.

**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?

**Teacher Reflection**

- Did this lesson wrap up the unit well? How else could you modify this lesson so that students see the relevance of isometric drawings?
- Were students easily able to recall terms such as mat plan? If not, how could you review the material?

### Using Cubes and Isometric Drawings

Explore polyhedra using different representations and perspectives for three dimensional block figures.

### Exploring the Isometric Drawing Tool

### Finding Surface Area and Volume

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

### 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.